91Ó°ÊÓ

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

Finding an Indefinite Integral In Exercises \(27-34,\) use substitution and partial fractions to find the indefinite integral. $$ \int \frac{e^{x}}{\left(e^{x}-1\right)\left(e^{x}+4\right)} d x $$

Finding an Indefinite Integral In Exercises \(27-34,\) use substitution and partial fractions to find the indefinite integral. $$ \int \frac{e^{x}}{\left(e^{2 x}+1\right)\left(e^{x}-1\right)} d x $$

Finding an Indefinite Integral In Exercises \(15-46\) , find the indefinite integral. $$ \int \csc ^{2} x e^{\cot x} d x $$

Finding an Indefinite Integral In Exercises \(21-36,\) find the indefinite integral. $$ \int \frac{\sqrt{1-x}}{\sqrt{x}} d x $$

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

Finding an Indefinite Integral Involving Secant and Tangent In Exercises \(19-32,\) find the indefinite integral. $$ \int \frac{\tan ^{2} x}{\sec ^{5} x} d x $$

Evaluating an Improper Integral In Exercises \(17-32\) , determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{\infty} \sin \frac{x}{2} d x $$

In Exercises 31–34, solve the differential equation. $$ y^{\prime}=\arctan \frac{x}{2} $$

Use integration tables to find the indefinite integral. \(\int \frac{e^{x}}{\left(1-e^{2 x}\right)^{3 / 2}} d x\)