8. Definite Integrals

66. Integrating derivatives

Use integration by parts to show that if f' is continuous on [a, b], then

∫[a to b] f(x)f'(x) dx = (1/2)[f(b)² - f(a)²]

Evaluate the indefinite integral.

${\int }^{}sin\theta {(1+\sin \theta )}^{84}\cos \theta d\theta$

Evaluating integrals Evaluate the following integrals.

∫₁ᵉ d𝓍 / [𝓍(1 + ln 𝓍)]

9–61. Trigonometric integrals Evaluate the following integrals.

40. ∫[0 to π/6] tan⁵(2x) sec(2x) dx

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)