91Ó°ÊÓ

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{0}^{\pi} \tan x d x\)

Show that \(\int(\ln x)^{n} d x=x(\ln x)^{n}-n \int(\ln x)^{n-1} d x\).

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int \frac{1}{\sqrt{9+x^{2}}} d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int x^{2} \sqrt{x^{2}-9} d x\)

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{0}^{\infty} \frac{x^{2}+3}{x+1} d x\)

Show that \(\int x^{n} e^{x} d x=x^{n} e^{x}-n \int x^{n-1} e^{x} d x\).

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{1}^{\infty} \frac{1}{(x+1)^{3}} d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int_{0}^{\frac{3}{4}} \sqrt{9-4 x^{2}} d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int \sqrt{4-9 x^{2}} d x\)

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{1}^{e^{2}} \frac{d x}{x \sqrt{\ln x}}\)