8. Definite Integrals

Evaluate the integrals in Exercises 53–76.

63. ∫(from -1 to -√2/2)dy/(y√(4y²-1))

7–64. Integration review Evaluate the following integrals.

44. ∫ from 0 to √3 of (6x³) / √(x² + 1) dx

Evaluate the definite integral.

${\int }_1^2(x-3){(x^2-6x)}^{2}dx$

Evaluate the indefinite integral.

${\int }_{}^{}\theta \bullet se{c}^2(5{\theta }^2+1)d\theta$

Evaluate the integrals in Exercises 33–52.

∫ from -π/4 to π/4 of 6 tan⁴(x) dx

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ dx / (4 - x²)^(3/2) from 0 to 1

Evaluating integrals Evaluate the following integrals.

∫₀^²π cos² 𝓍/6 d𝓍

Evaluate the integrals in Exercises 1–22.

∫₀^π 8 sin⁴(y) cos²(y) dy

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

 ∫₀^π/⁴ eˢᶦⁿ² ˣ sin 2𝓍 d𝓍

Evaluate the integrals in Exercises 47–68.

∫₀¹/² x³ (1 + 9x⁴)⁻³/² 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)

7–64. Integration review Evaluate the following integrals.

51. ∫ from -1 to 0 of x / (x² + 2x + 2) dx

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

∫₀^{π} 2^{sin x} · cos x dx

7–84. Evaluate the following integrals.

30. ∫ from 5/2 to 5√3/2 [1 / (v² √(25 - v²))] dv

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  ∑ (2cₖ - 1)⁻¹/² ∆xₖ, where P is a partition of [1, 5]

 ∥P∥→0  k = 1