91Ó°ÊÓ

Evaluate the integral. $$ \int x \sqrt[3]{x+c} d x $$

Find the values of \(p\) for which the integral converges and evaluate the integral for those values of \(p .\) $$ \int_{0}^{1} \frac{1}{x^{p}} d x $$

Evaluate the integral. $$ \int \frac{x \ln x}{\sqrt{x^{2}-1}} d x $$

Find the values of \(p\) for which the integral converges and evaluate the integral for those values of \(p .\) $$ \int_{e}^{\infty} \frac{1}{x(\ln x)^{p}} d x $$

Find the area of the region bounded by the given curves. $$ y=x^{2} e^{-x}, \quad y=x e^{-x} $$

Evaluate the integral. $$ \int \frac{d x}{x^{4}-16} $$

Find the values of \(p\) for which the integral converges and evaluate the integral for those values of \(p .\) $$ \int_{0}^{1} x^{p} \ln x d x $$

The German mathematician Karl Weierstrass (1815-1897) noticed that the substitution t=\tan (x / 2) will convert any rational function of \(\sin x\) and \(\cos x\) into an ordinary rational function of t . $$ \begin{array}{l}{\text { (a) If } t=\tan (x / 2),-\pi

(a) Evaluate the integral \(\int_{0}^{\infty} x^{n} e^{-x} d x\) for \(n=0,1,2,\) and \(3 .\) (b) Guess the value of \(\int_{0}^{\infty} x^{n} e^{-x} d x\) when \(n\) is an arbitrary positive integer. (c) Prove your guess using mathematical induction.

Use a graph of the integrand to guess the value of the integral. Then use the methods of this section to prove that your guess is correct. $$ \int_{0}^{2} \sin 2 \pi x \cos 5 \pi x d x $$