91Ó°ÊÓ

Evaluating an Improper Integral In Exercises \(17-32\) , determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{0} x e^{-4 x} d x $$

Evaluating a Limit In Exercises \(11-42,\) evaluate the limit, using L'Hopital's Rule if necessary. $$ \lim _{x \rightarrow 0} \frac{\arcsin x}{x} $$

Finding an Indefinite Integral In Exercises \(15-46\) , find the indefinite integral. $$ \int \frac{t^{2}-3}{-t^{3}+9 t+1} d t $$

Finding an Indefinite Integral In Exercises \(21-36,\) find the indefinite integral. $$ \int \frac{x^{2}}{\sqrt{36-x^{2}}} d x $$

In Exercises 11–30, find the indefinite integral. (Note: Solve by the simplest method—not all require integration by parts.) $$ \int \frac{x}{\sqrt{6 x+1}} d x $$

Evaluating an Improper Integral In Exercises \(17-32\) , determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{\infty} x e^{-x / 3} d x $$

Evaluating a Limit In Exercises \(11-42,\) evaluate the limit, using L'Hopital's Rule if necessary. $$ \lim _{x \rightarrow 1} \frac{\arctan x-(\pi / 4)}{x-1} $$

Use integration tables to find the indefinite integral. \(\int \frac{\theta^{3}}{1+\sin \theta^{4}} d \theta\)

Finding an Indefinite Integral In Exercises \(15-46\) , find the indefinite integral. $$ \int \frac{x+1}{\sqrt{3 x^{2}+6 x}} d x $$

Finding an Indefinite Integral Involving Secant and Tangent In Exercises \(19-32,\) find the indefinite integral. $$ \int \tan ^{6} 3 x d x $$