91Ó°ÊÓ

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{0}^{3} \frac{x}{x^{2}-2} d x\)

Integrate by parts successively to evaluate the given indefinite integral. $$ \int x^{3} e^{x} d x $$

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{-\infty}^{0} \frac{1}{\left(1+x^{2}\right)^{3 / 2}} d x $$

Integrate by parts successively to evaluate the given indefinite integral. $$ \int x^{3} \sin (x) d x $$

Use the method of partial fractions to calculate the given integral. $$ \int \frac{3 x+4}{x^{2}+x-6} d x $$

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int_{\sqrt{3}}^{\sqrt{6}} \frac{2}{\left(x^{2}-2\right)^{3 / 2}} d x $$

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{-\infty}^{0} \frac{1}{\left(1+x^{2}\right)^{2}} d x $$

Use the method of partial fractions to decompose the integrand. Then evaluate the given integral. \(\int \frac{2 x^{2}+4 x+2}{\left(x^{2}+1\right)^{3}} d x\)

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{-\infty}^{4} e^{x / 3} d x $$

Use the method of partial fractions to calculate the given integral. $$ \int \frac{9 x+18}{(x-3)(x+6)} d x $$