12. Techniques of Integration

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

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 ${\int }_3^{k}f\left(x\right) dx$?

For the integral ${\int }_{}^{}x{e}^{x}dx$, which of the following correctly identifies $u$ and $du$ for use in integration by parts?

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

60. ∫ x² coshx dx

Finding volume

Let R be the "triangular" region in the first quadrant that is bounded above by the line y = 1, below by the curve y = ln x, and on the left by the line x = 1.

Find the volume of the solid generated by revolving R about

a. the x-axis.

2. When applying the formula for integration by parts, how do you choose the u and dv? How can you apply integration by parts to an integral of the form ∫ f(x) dx?

Find the integral.

${\int }_{}^{}s{e}^{3s}ds$

Find the integral.

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

Find the integral.

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

Find the indefinite integral.

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

Find the indefinite integral.

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

Evaluate the indefinite integral.

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

Evaluate the indefinite integral.

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