8. Definite Integrals

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)

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

57. ∫ (from 0 to √3/2) 4/(9 + 4x²) dx

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    

  ∫₀^π/² (cos θ sin θ) / √(cos² θ + 16) dθ (Hint: Begin with u = cos θ .)

Change of variables Use the change of variables u³ = 𝓍² ― 1 to evaluate the integral ∫₁³ 𝓍∛(𝓍²―1) d𝓍 .

On which derivative rule is the Substitution Rule based?

The composite function ƒ(g(𝓍)) consists of an inner function g and an outer function ƒ. If an integrand includes ƒ(g(𝓍)), which function is often a likely choice for a new variable u?

When using a change of variables u = g(𝓍) to evaluate the definite integral ∫ₐᵇ ƒ(g(𝓍)) g' (𝓍) d(𝓍), how are the limits of integration transformed?

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫₀³ (2x - 1) / (x + 1) dx