8. Definite Integrals

Evaluating integrals Evaluate the following integrals.

∫₁ᵉ d𝓍 / [𝓍(1 + ln 𝓍)]

Evaluating integrals Evaluate the following integrals.

∫₀¹ 𝓍 • 2ˣ²⁺¹ d𝓍

Evaluating integrals Evaluate the following integrals.

∫₀² (2𝓍 + 1)³ d𝓍

Evaluating integrals Evaluate the following integrals.

∫₀^²π cos² 𝓍/6 d𝓍

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             

 ∫ 𝓍 cos²𝓍² d𝓍

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.                                                                                                                                                           

(a) ∫ ƒ(𝓍) ƒ'(𝓍) d𝓍 = ½ (ƒ(𝓍))² + C.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.                                                                                                                                                           

(b) ∫ (ƒ(𝓍))ⁿ ƒ'(𝓍) d𝓍 = 1/(n + 1) (ƒ(𝓍))ⁿ⁺¹ + C , n ≠ ―1 .

2. What change of variables is suggested by an integral containing √(x² + 36)?

6. Using the trigonometric substitution x = 8 sec θ, where x ≥ 8 and 0 < θ ≤ π/2, express tan θ in terms of x.

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

9. ∫[5 to 5√3] √(100 - x²) dx

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

12. ∫[1/2 to 1] √(1 - x²)/x² dx

Evaluating integrals Evaluate the following integrals.

∫π/₁₂^π/⁹ (csc 3𝓍 cot 3𝓍 + sec 3𝓍 tan 3𝓍) d𝓍

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

 ∫₁³ ( 2ˣ / 2ˣ + 4 ) d𝓍

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

18. ∫ (from 0 to √2) (x + 1)/(3x² + 6) dx

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

25. ∫ (from -3/2 to -1) dx/(4x² + 12x + 10)