8. Definite Integrals

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

 ∫₁/₃^¹/√³ 4/(9𝓍² + 1) d𝓍

9–61. Trigonometric integrals Evaluate the following integrals.

53. ∫ from 0 to π/4 of sec⁴θ dθ

Evaluating integrals Evaluate the following integrals.

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

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

12. ∫[1/2 to 1] √(1 - x²)/x² dx

Evaluate the integrals in Exercises 23–32.

∫_{π/2}^{3π/4} √(1 - sin(2x)) dx

On which derivative rule is the Substitution Rule based?

Evaluate the indefinite integral.

${\int }_{}^{}\frac{x}{{(x-6)}^5}dx$

Evaluate the integrals in Exercises 67–74 in terms of

b. natural logarithms.

67. ∫(from 0 to 2√3)dx/√(4+x²)

Evaluating integrals Evaluate the following integrals.

∫₀² (2𝓍 + 1)³ d𝓍

Evaluate the integrals in Exercises 47–68.

∫₀^π/4 sec²x / (1 + 7 tan x)²/³ dx

7–64. Integration review Evaluate the following integrals.

18. ∫ from 3 to 7 of (t - 6) * √(t - 3) dt

Evaluate the integrals in Exercises 47–68.

∫₀ ^π tan² (θ/3) dθ

9–61. Trigonometric integrals Evaluate the following integrals.

22. ∫[π/4 to π/2] sin²(2x) cos³(2x) dx

7–84. Evaluate the following integrals.

35. ∫ from 0 to π/4 [(tan²θ + tanθ + 1) sec²θ] dθ

Evaluate the indefinite integral.

${\int }_{}^{}3t\sqrt{t^2+7}dt$