12. Techniques of Integration

Find the integral.

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

Evaluate the indefinite integral.

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

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: ${\int }_{}^{}k{x}^{n-1}dx$ = $\frac{1}{n}$?

Evaluate the indefinite integral.

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

Find the indefinite integral.

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

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

Find the indefinite integral.

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

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

Evaluate the definite integral.

${\int }_0^{\pi }{t}^2\sin tdt$

Evaluate the indefinite integral.

${\int }_{}^{}{x}^2\cos 4xdx$

Find the integral.

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

Evaluate the definite integral.

${\int }_1^{2}x\ln xdx$