12. Techniques of Integration

Find the integral.

${\int }_{}^{}{x}^2\ln xdx$

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ x·sec²x dx

Evaluate the indefinite integral.

${\int }_{}^{}(s^2+4)e^{3s}ds$

Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^(-x), and the line x = 1.

a. About the y-axis.

1. On which derivative rule is integration by parts based?

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

∫ √(x - 2) / √(x - 1) dx

Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.

∫ z(ln z)² dz

Given that the definite integral from $0$ to $\pi$ of ${\sin }^4\left(x\right)dx$ equals $\frac{3}{8\pi }$, what is the value of the definite integral from $0$ to $\pi$ of ${\cos }^4\left(x\right)dx$?

If n is a known positive integer, for what value of k does the following hold: ${\int }_{}^{}k{x}^{n-1}dx$ = $\frac{1}{n}$?

49. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample:

c. ∫ v du = u·v - ∫ u dv

Evaluate the limits in Exercise 9 and 10 by identifying them with definite integrals and evaluating the integrals.

lim (n → ∞) Σ (from k=1 to n) ln √(1 + k/n)

Practice with tabular integration Evaluate the following integrals using tabular integration (refer to Exercise 77).

b. ∫ 7x e³ˣ dx

Verify the integration formulas in Exercises 111–114.

113. ∫ (arcsin x)² dx = x(arcsin x)² - 2x + 2 √(1 - x²) arcsin x + C

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

∫ (xe^x) / (x + 1)² dx

Use any method to evaluate the integrals in Exercises 65–70.

∫ x cos³(x) dx