{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1. Use integration by parts to evaluate the definite integral "},{"type":"inlineMath","attrs":{"latex":"\\int_{0}^{\\frac{\\pi}{4}} x^3 \\cos(2x) \\, dx"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Integration by Parts (IBP)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to apply the integration by parts technique to a definite integral involving a product of a polynomial and a trigonometric function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Integration by Parts formula: "},{"type":"inlineMath","attrs":{"latex":"\\int u \\, dv = u v - \\int v \\, du"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"LIATE rule: Used to choose "},{"type":"inlineMath","attrs":{"latex":"u"}},{"type":"text","text":" (Logarithmic, Inverse trig, Algebraic, Trig, Exponential)"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify "},{"type":"inlineMath","attrs":{"latex":"u"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"dv"}},{"type":"text","text":" using LIATE. Here, "},{"type":"inlineMath","attrs":{"latex":"u = x^3"}},{"type":"text","text":" (algebraic) and "},{"type":"inlineMath","attrs":{"latex":"dv = \\cos(2x) \\, dx"}},{"type":"text","text":" (trig)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compute "},{"type":"inlineMath","attrs":{"latex":"du"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"v"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"du = 3x^2 \\, dx"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"v = \\int \\cos(2x) \\, dx = \\frac{1}{2} \\sin(2x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the IBP formula: "},{"type":"inlineMath","attrs":{"latex":"\\int u \\, dv = u v - \\int v \\, du"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the new integral: "},{"type":"inlineMath","attrs":{"latex":"\\int_{0}^{\\frac{\\pi}{4}} x^3 \\cos(2x) \\, dx = [x^3 \\cdot \\frac{1}{2} \\sin(2x)]_{0}^{\\frac{\\pi}{4}} - \\int_{0}^{\\frac{\\pi}{4}} \\frac{1}{2} \\sin(2x) \\cdot 3x^2 \\, dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Now, focus on the remaining integral "},{"type":"inlineMath","attrs":{"latex":"\\int x^2 \\sin(2x) \\, dx"}},{"type":"text","text":" and consider applying IBP again."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\int_{0}^{\\frac{\\pi}{4}} x^3 \\cos(2x) \\, dx = \\frac{3\\pi^2}{128}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The integral requires repeated application of integration by parts. After evaluating and plugging in the limits, you arrive at the final value."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Use integration by parts to evaluate the definite integral "},{"type":"inlineMath","attrs":{"latex":"\\int_{1}^{e} (\\ln x)^2 \\, dx"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Integration by Parts (IBP)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use IBP for an integral involving a logarithmic function squared."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Integration by Parts formula: "},{"type":"inlineMath","attrs":{"latex":"\\int u \\, dv = u v - \\int v \\, du"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Choose "},{"type":"inlineMath","attrs":{"latex":"u = (\\ln x)^2"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"dv = dx"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Let "},{"type":"inlineMath","attrs":{"latex":"u = (\\ln x)^2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"dv = dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compute "},{"type":"inlineMath","attrs":{"latex":"du = 2 \\ln x \\cdot \\frac{1}{x} dx"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"v = x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply IBP: "},{"type":"inlineMath","attrs":{"latex":"\\int u \\, dv = u v - \\int v \\, du"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up: "},{"type":"inlineMath","attrs":{"latex":"\\int_{1}^{e} (\\ln x)^2 dx = [x (\\ln x)^2]_{1}^{e} - \\int_{1}^{e} x \\cdot 2 \\ln x \\cdot \\frac{1}{x} dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the remaining integral to "},{"type":"inlineMath","attrs":{"latex":"2 \\int_{1}^{e} \\ln x \\, dx"}},{"type":"text","text":" and prepare to evaluate."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\int_{1}^{e} (\\ln x)^2 dx = e - 2"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After applying IBP and evaluating the resulting integrals, you find the value is "},{"type":"inlineMath","attrs":{"latex":"e - 2"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Evaluate the integral "},{"type":"inlineMath","attrs":{"latex":"\\int \\sec^3 x \\tan^3 x \\, dx"}},{"type":"text","marks":[{"type":"bold"}],"text":" using substitution, IBP, and/or trig identities."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Trigonometric Integrals"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to manipulate powers of secant and tangent using trigonometric identities and substitution."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Trig identities: "},{"type":"inlineMath","attrs":{"latex":"\\tan^2 x = \\sec^2 x - 1"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitution: Let "},{"type":"inlineMath","attrs":{"latex":"u = \\tan x"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"u = \\sec x"}},{"type":"text","text":" as appropriate"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite "},{"type":"inlineMath","attrs":{"latex":"\\tan^3 x"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"\\tan x \\cdot \\tan^2 x"}},{"type":"text","text":" and use the identity "},{"type":"inlineMath","attrs":{"latex":"\\tan^2 x = \\sec^2 x - 1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the integrand in terms of "},{"type":"inlineMath","attrs":{"latex":"\\tan x"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"\\sec x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Consider substitution: Let "},{"type":"inlineMath","attrs":{"latex":"u = \\tan x"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"du = \\sec^2 x dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Alternatively, expand and split the integral into simpler parts."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the resulting integrals for evaluation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\int \\sec^3 x \\tan^3 x \\, dx = \\frac{1}{4} \\sec^4 x - \\frac{1}{2} \\sec^2 x + C"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"By rewriting and substituting, the integral simplifies to a combination of secant powers."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. Evaluate the definite integral "},{"type":"inlineMath","attrs":{"latex":"\\int_{0}^{\\frac{\\pi}{2}} \\sin^3 x \\cos^2 x \\, dx"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Trigonometric Integrals"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use trig identities and substitution to evaluate definite integrals involving powers of sine and cosine."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Trig identities: "},{"type":"inlineMath","attrs":{"latex":"\\sin^2 x = \\frac{1}{2}(1 - \\cos 2x)"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\cos^2 x = \\frac{1}{2}(1 + \\cos 2x)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitution: Let "},{"type":"inlineMath","attrs":{"latex":"u = \\sin x"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"u = \\cos x"}},{"type":"text","text":" as appropriate"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite "},{"type":"inlineMath","attrs":{"latex":"\\sin^3 x"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"\\sin x \\cdot \\sin^2 x"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"\\cos^2 x"}},{"type":"text","text":" using the double angle identity."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expand the product and express in terms of "},{"type":"inlineMath","attrs":{"latex":"\\sin x"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"\\cos 2x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Consider substitution or further simplification using identities."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the resulting integrals for evaluation over "},{"type":"inlineMath","attrs":{"latex":"[0, \\frac{\\pi}{2}]"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\int_{0}^{\\frac{\\pi}{2}} \\sin^3 x \\cos^2 x \\, dx = \\frac{4}{15}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Using trig identities and integrating term by term, the definite integral evaluates to "},{"type":"inlineMath","attrs":{"latex":"\\frac{4}{15}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5. Use trigonometric substitution to evaluate the definite integral "},{"type":"inlineMath","attrs":{"latex":"\\int_{0}^{3} \\sqrt{9 - x^2} \\, dx"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Trigonometric Substitution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use trig substitution for integrals involving square roots of quadratic expressions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"a^2 - x^2"}},{"type":"text","text":", use "},{"type":"inlineMath","attrs":{"latex":"x = a \\sin \\theta"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identity: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{a^2 - x^2} = a \\cos \\theta"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Let "},{"type":"inlineMath","attrs":{"latex":"x = 3 \\sin \\theta"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"dx = 3 \\cos \\theta d\\theta"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Change the limits: When "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\theta = 0"}},{"type":"text","text":"; when "},{"type":"inlineMath","attrs":{"latex":"x = 3"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\theta = \\frac{\\pi}{2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute into the integral: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{9 - x^2} = 3 \\cos \\theta"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the new integral in terms of "},{"type":"inlineMath","attrs":{"latex":"\\theta"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\int_{0}^{3} \\sqrt{9 - x^2} \\, dx = \\frac{9\\pi}{4}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This is the area of a quarter circle of radius 3, which matches the result from trig substitution."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. Use trigonometric substitution to evaluate the integral "},{"type":"inlineMath","attrs":{"latex":"\\int \\frac{1}{x^2 + 6x + 13} \\, dx"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Trigonometric Substitution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use trig substitution for integrals involving quadratic denominators."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Complete the square: "},{"type":"inlineMath","attrs":{"latex":"x^2 + 6x + 13 = (x + 3)^2 + 4"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"a^2 + u^2"}},{"type":"text","text":", use "},{"type":"inlineMath","attrs":{"latex":"u = a \\tan \\theta"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identity: "},{"type":"inlineMath","attrs":{"latex":"\\int \\frac{1}{a^2 + x^2} dx = \\frac{1}{a} \\arctan \\left(\\frac{x}{a}\\right) + C"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Complete the square in the denominator: "},{"type":"inlineMath","attrs":{"latex":"x^2 + 6x + 13 = (x + 3)^2 + 4"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Let "},{"type":"inlineMath","attrs":{"latex":"u = x + 3"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"du = dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite the integral as "},{"type":"inlineMath","attrs":{"latex":"\\int \\frac{1}{u^2 + 2^2} du"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the standard arctangent integral formula."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\int \\frac{1}{x^2 + 6x + 13} dx = \\frac{1}{2} \\arctan \\left(\\frac{x + 3}{2}\\right) + C"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Completing the square and using the arctangent formula gives the result."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7. Use trigonometric substitution to evaluate the integral "},{"type":"inlineMath","attrs":{"latex":"\\int \\frac{x}{\\sqrt{x^2 + 10x + 9}} \\, dx"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Trigonometric Substitution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use trig substitution for integrals involving a linear numerator and a square root denominator."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Complete the square: "},{"type":"inlineMath","attrs":{"latex":"x^2 + 10x + 9 = (x + 5)^2 - 16"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"u^2 - a^2"}},{"type":"text","text":", use "},{"type":"inlineMath","attrs":{"latex":"u = a \\sec \\theta"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Complete the square: "},{"type":"inlineMath","attrs":{"latex":"x^2 + 10x + 9 = (x + 5)^2 - 16"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Let "},{"type":"inlineMath","attrs":{"latex":"u = x + 5"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"du = dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite the integral as "},{"type":"inlineMath","attrs":{"latex":"\\int \\frac{u - 5}{\\sqrt{u^2 - 16}} du"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Consider substitution "},{"type":"inlineMath","attrs":{"latex":"u = 4 \\sec \\theta"}},{"type":"text","text":" and rewrite the integral in terms of "},{"type":"inlineMath","attrs":{"latex":"\\theta"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\int \\frac{x}{\\sqrt{x^2 + 10x + 9}} dx = \\sqrt{x^2 + 10x + 9} - 5 \\ln \\left| x + 5 + \\sqrt{x^2 + 10x + 9} \\right| + C"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After substitution and simplification, the integral resolves to this expression."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8. Use partial fraction decomposition to evaluate the integral "},{"type":"inlineMath","attrs":{"latex":"\\int \\frac{x}{x^2 + 10x + 9} dx"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Partial Fraction Decomposition"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to decompose rational functions and integrate each part."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor denominator: "},{"type":"inlineMath","attrs":{"latex":"x^2 + 10x + 9 = (x + 1)(x + 9)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up: "},{"type":"inlineMath","attrs":{"latex":"\\frac{x}{(x + 1)(x + 9)} = \\frac{A}{x + 1} + \\frac{B}{x + 9}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor the denominator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the partial fractions and solve for "},{"type":"inlineMath","attrs":{"latex":"A"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"B"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the integral as the sum of two simpler integrals."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Integrate each term separately."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\int \\frac{x}{x^2 + 10x + 9} dx = \\frac{1}{8} \\ln |x + 1| + \\frac{7}{8} \\ln |x + 9| + C"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After decomposing and integrating, you get a sum of logarithms."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9. Use partial fraction decomposition to evaluate the integral "},{"type":"inlineMath","attrs":{"latex":"\\int \\frac{x^2 + x + 2}{(x + 1)(x^2 + 1)} dx"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Partial Fraction Decomposition"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to decompose a rational function with a quadratic denominator and integrate each part."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up: "},{"type":"inlineMath","attrs":{"latex":"\\frac{x^2 + x + 2}{(x + 1)(x^2 + 1)} = \\frac{A}{x + 1} + \\frac{Bx + C}{x^2 + 1}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the partial fraction decomposition."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply both sides by the denominator and equate coefficients to solve for "},{"type":"inlineMath","attrs":{"latex":"A"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"B"}},{"type":"text","text":", and "},{"type":"inlineMath","attrs":{"latex":"C"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the integral as the sum of three simpler integrals."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Integrate each term separately."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\int \\frac{x^2 + x + 2}{(x + 1)(x^2 + 1)} dx = \\ln|x + 1| + \\frac{1}{2} \\ln(x^2 + 1) + \\arctan x + C"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After decomposing and integrating, you get a combination of logarithmic and arctangent terms."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q10. Use integration techniques to evaluate the integral "},{"type":"inlineMath","attrs":{"latex":"\\int \\frac{\\cos^3(3x)}{\\sqrt{\\sin(3x)}} dx"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Mixed Integration Techniques"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to combine substitution and trigonometric identities for integrals involving powers and roots."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitution: Let "},{"type":"inlineMath","attrs":{"latex":"u = \\sin(3x)"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"du = 3 \\cos(3x) dx"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Trig identities for powers of cosine"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Let "},{"type":"inlineMath","attrs":{"latex":"u = \\sin(3x)"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"du = 3 \\cos(3x) dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express "},{"type":"inlineMath","attrs":{"latex":"\\cos^3(3x)"}},{"type":"text","text":" in terms of "},{"type":"inlineMath","attrs":{"latex":"u"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"du"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite the integral in terms of "},{"type":"inlineMath","attrs":{"latex":"u"}},{"type":"text","text":" and simplify."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the resulting integral for evaluation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\int \\frac{\\cos^3(3x)}{\\sqrt{\\sin(3x)}} dx = \\frac{2}{9} (1 - \\sin(3x))^{3/2} + C"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitution and simplification lead to this result."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q11. Use integration techniques to evaluate the integral "},{"type":"inlineMath","attrs":{"latex":"\\int \\frac{x}{\\sqrt{x^2 + 4x + 8}} dx"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Mixed Integration Techniques"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use substitution for integrals involving a linear numerator and a square root denominator."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Complete the square: "},{"type":"inlineMath","attrs":{"latex":"x^2 + 4x + 8 = (x + 2)^2 + 4"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitution: Let "},{"type":"inlineMath","attrs":{"latex":"u = x + 2"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Complete the square in the denominator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Let "},{"type":"inlineMath","attrs":{"latex":"u = x + 2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"du = dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite the integral as "},{"type":"inlineMath","attrs":{"latex":"\\int \\frac{u - 2}{\\sqrt{u^2 + 4}} du"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Split the integral and set up for evaluation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\int \\frac{x}{\\sqrt{x^2 + 4x + 8}} dx = \\sqrt{x^2 + 4x + 8} - 2 \\ln \\left| x + 2 + \\sqrt{x^2 + 4x + 8} \\right| + C"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitution and splitting the integral lead to this result."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q12. Use integration techniques to evaluate the definite integral "},{"type":"inlineMath","attrs":{"latex":"\\int_{0}^{\\frac{\\pi}{4}} \\tan^3 x \\sec x dx"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Mixed Integration Techniques"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to integrate powers of tangent and secant using substitution and identities."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Trig identity: "},{"type":"inlineMath","attrs":{"latex":"\\tan^2 x = \\sec^2 x - 1"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitution: Let "},{"type":"inlineMath","attrs":{"latex":"u = \\tan x"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"du = \\sec^2 x dx"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite "},{"type":"inlineMath","attrs":{"latex":"\\tan^3 x \\sec x"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"\\tan x \\tan^2 x \\sec x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the identity "},{"type":"inlineMath","attrs":{"latex":"\\tan^2 x = \\sec^2 x - 1"}},{"type":"text","text":" to expand."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the integrand in terms of "},{"type":"inlineMath","attrs":{"latex":"\\tan x"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"\\sec x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the integral for substitution."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\int_{0}^{\\frac{\\pi}{4}} \\tan^3 x \\sec x dx = \\frac{1}{4}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Using identities and substitution, the definite integral evaluates to "},{"type":"inlineMath","attrs":{"latex":"\\frac{1}{4}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q13. Use integration techniques to evaluate the definite integral "},{"type":"inlineMath","attrs":{"latex":"\\int_{-1}^{2} \\frac{x}{x^2 + 2x + 10} dx"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Mixed Integration Techniques"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use substitution for definite integrals involving a linear numerator and a quadratic denominator."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Complete the square: "},{"type":"inlineMath","attrs":{"latex":"x^2 + 2x + 10 = (x + 1)^2 + 9"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitution: Let "},{"type":"inlineMath","attrs":{"latex":"u = x + 1"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Complete the square in the denominator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Let "},{"type":"inlineMath","attrs":{"latex":"u = x + 1"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"du = dx"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite the integral as "},{"type":"inlineMath","attrs":{"latex":"\\int_{u_1}^{u_2} \\frac{u - 1}{u^2 + 9} du"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Split the integral and set up for evaluation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\int_{-1}^{2} \\frac{x}{x^2 + 2x + 10} dx = \\frac{1}{2} \\ln \\left( \\frac{13}{10} \\right) - \\frac{1}{3} \\left[ \\arctan \\left( \\frac{3}{9} \\right) - \\arctan \\left( \\frac{-2}{9} \\right) \\right]"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitution and splitting the integral lead to this result."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q14. 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Use integration techniques to evaluate the integral "},{"type":"inlineMath","attrs":{"latex":"\\int \\frac{x^2 + 8x - 3}{x^3 + 3x^2} dx"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Partial Fraction Decomposition"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to decompose and integrate rational functions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor denominator: "},{"type":"inlineMath","attrs":{"latex":"x^3 + 3x^2 = x^2(x + 3)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up: "},{"type":"inlineMath","attrs":{"latex":"\\frac{x^2 + 8x - 3}{x^3 + 3x^2} = \\frac{A}{x} + \\frac{B}{x^2} + \\frac{C}{x + 3}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor the denominator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the partial fraction decomposition and solve for "},{"type":"inlineMath","attrs":{"latex":"A"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"B"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"C"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the integral as the sum of three simpler integrals."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Integrate each term separately."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\int \\frac{x^2 + 8x - 3}{x^3 + 3x^2} dx = \\ln|x| + \\frac{8}{x} - \\ln|x + 3| + C"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Decomposing and integrating gives the result."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q20. 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