91Ó°ÊÓ

Make a substitution to express the integrand as a rational function and then evaluate the integral. $$ \int \frac{\cosh t}{\sinh ^{2} t+\sinh ^{4} t} d t $$

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking C = 0). \(\int \sin ^{5} x \cos ^{3} x d x\)

Use integration by parts to prove the reduction formula. $$ \int x^{n} e^{x} d x=x^{n} e^{x}-n \int x^{n-1} e^{x} d x $$

Evaluate the integral. $$ \int \frac{d x}{x\left(x^{4}+1\right)} $$

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking C = 0). \(\int \sin ^{5} x \cos ^{3} x d x\)

Evaluate the integral. $$ \int x^{2} \sinh m x d x $$

Use integration by parts, together with the techniques of this section, to evaluate the integral. $$ \int \ln \left(x^{2}-x+2\right) d x $$

Use the Comparison Theorem to determine whether the integral is convergent or divergent. $$ \int_{0}^{\pi} \frac{\sin ^{2} x}{\sqrt{x}} d x $$

Use integration by parts, together with the techniques of this section, to evaluate the integral. $$ \int x \tan ^{-1} x d x $$

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking C = 0). \(\int \sec ^{4}\left(\frac{1}{2} x\right) d x\)