91Ó°ÊÓ

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{3}^{6}(x-4)^{-2} d x\)

Use Heaviside's method to calculate the partial fraction decomposition of the given rational function. $$ \frac{2 x^{2}+8 x+13}{(x+1)(x+3)(x+5)} $$

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{-\infty}^{2} \frac{1}{\sqrt{3-x}} d x $$

Integrate by parts successively to evaluate the given indefinite integral. $$ \int x^{2} \sin (x) d x $$

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

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

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int_{\sqrt{2}}^{\sqrt{5}} \frac{2 x^{2}}{\sqrt{x^{2}-1}} d x $$

Integrate by parts successively to evaluate the given indefinite integral. $$ \int 4 x^{2} \cos (2 x) d x $$

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

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{-2}^{4}(x+1)^{-2 / 3} d x\)