91Ó°ÊÓ

Use a table of integrals with forms involving \(e^{M}\) to find the integral. $$ \int \frac{1}{1+e^{2 x}} d x $$

Evaluate the limit, using L'Hôpital's Rule if necessary. (In Exercise \(18, n\) is a positive integer.) $$ \lim _{x \rightarrow 2} \frac{x^{2}-x-2}{x-2} $$

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int x e^{-2 x} d x $$

Explain why the evaluation of the integral is incorrect. Use the integration capabilities of a graphing utility to attempt to evaluate the integral. Determine whether the utility gives the correct answer. $$ \int \frac{1}{x^{2}} d x=-2 $$

Use partial fractions to find the integral.\(\int \frac{5-x}{2 x^{2}+x-1} d x\)

Find the indefinite integral using the substitution \(x=2 \sec \theta\). $$ \int x^{3} \sqrt{x^{2}-4} d x $$

Evaluate the limit, using L'Hôpital's Rule if necessary. (In Exercise \(18, n\) is a positive integer.) $$ \lim _{x \rightarrow-1} \frac{x^{2}-x-2}{x+1} $$

Use partial fractions to find the integral.\(\int \frac{5 x^{2}-12 x-12}{x^{3}-4 x} d x\)

Find the indefinite integral using the substitution \(x=2 \sec \theta\). $$ \int \frac{x^{3}}{\sqrt{x^{2}-4}} d x $$

Explain why the evaluation of the integral is incorrect. Use the integration capabilities of a graphing utility to attempt to evaluate the integral. Determine whether the utility gives the correct answer. $$ \int_{2}^{2} \frac{-2}{(x-1)^{3}} d x=\frac{8}{9} $$