12. Techniques of Integration

Evaluate the indefinite integral.

${\int }^{}\frac{\ln x}{x^4}dx$

9–40. Integration by parts Evaluate the following integrals using integration by parts.

38. ∫ x² ln²(x) dx

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ arcsin(y) dy

9–40. Integration by parts Evaluate the following integrals using integration by parts.

20. ∫ sin⁻¹(x) dx

Find the indefinite integral.

${\int }_{}^{}{r}^2{e}^{-r}dr$

50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:

53. ∫ lnⁿ(x) dx = x lnⁿ(x) - n ∫ lnⁿ⁻¹(x) dx

Find the integral.

${\int }_{}^{}\theta \cos 2\theta d\theta$

Given that ${\int }_1^{3}f\left(x\right) dx$=$-3$ and ${\int }_k^{1}f\left(x\right) dx$=$4$, what is the value of ${\int }_3^{k}f\left(x\right) dx$?

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫₀^π/2 x³ cos 2x dx

Evaluate ∫ x³ √(1 - x²) dx using:

a. Integration by parts.

9–40. Integration by parts Evaluate the following integrals using integration by parts.

23. ∫ x² sin(2x) dx

60. Two Methods

a. Evaluate ∫(x · ln(x²)) dx using the substitution u = x² and evaluating ∫(ln(u)) du.

Find the indefinite integral.

${\int }_{}^{}{\theta }^2\cos 3\theta d\theta$

48. Integral of sec³x Use integration by parts to show that:

∫ sec³x dx = (1/2) secx tanx + (1/2) ∫ secx dx

18. Finding volume (Continuation of Exercise 17.) Find the volume of the solid generated by revolving the region R about:

a. the y-axis.