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 definite integral.

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

Evaluate the definite integral.

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

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 ${\cos }^4\left(x\right)dx$?

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?

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

28. ∫ ln² x dx

71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.

74. ∫xⁿ arcsin(x) dx (Hint: integration by parts.)

7–84. Evaluate the following integrals.

38. ∫ from π/6 to π/2 [cos x · ln(sin x)] dx