8. Definite Integrals

Evaluate the integrals in Exercises 39–56.

43. ∫(from 0 to π)(sin t)/(2 - cos t) dt

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

Evaluate the integrals in Exercises 39–56.

47. ∫(from 2 to 4)dx/(x(ln x)²)

Evaluate the integrals in Exercises 41–60.

55. ∫(from -π/4 to π/4)cosh(tanθ)sec²θ dθ

Evaluate the integrals in Exercises 23–32.

∫₀^(π/6) √(1 + sin(x)) dx

(Hint: Multiply by √((1 - sin(x)) / (1 - sin(x))))

Evaluate the indefinite integral.

${\int }_{}^{}\frac{1}{{(3x+2)}^5}dx$

Definite Integrals

In Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval, and the numbers cₖ are chosen from the subintervals of P.

     n

 lim  ∑ (cos(cₖ/2)) ∆xₖ, where P is a partition of [-π, 0]

 ∥P∥→0  k = 1

Evaluate the integrals in Exercises 67–74 in terms of

b. natural logarithms.

69. ∫(from 5/4 to 2)dx/(1-x²)

Evaluate the integrals in Exercises 23–32.

∫₋π^π (1 - cos²(t))^(3/2) dt

Evaluate the definite integral.

${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\tan y$ $se{c}^{2}ydy$

What is the largest value that

∫ from a to b x√(2x - x²) dx

can have for any a and b? Give reasons for your answer.

9–61. Trigonometric integrals Evaluate the following integrals.

57. ∫ from 0 to π of (1 - cos2x)³ᐟ² 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

7–84. Evaluate the following integrals.

11. ∫ from 0 to π/4 (sec x – cos x)² dx

7–64. Integration review Evaluate the following integrals.

16. ∫ from 0 to 1 of (t² / (1 + t⁶)) dt