91Ó°ÊÓ

Decompose the given rational function into partial fractions. Calculate the coefficients. $$ \frac{36}{(x-4)^{2}(x+2)} $$

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{4}^{13} \frac{1}{\sqrt{x-4}} d x\)

Explicitly calculate the partial fraction decomposition of the given rational function. \(\frac{x^{3}+12 x^{2}-9 x+48}{(x-3)\left(x^{2}+4\right)}\)

Integrate by parts to evaluate the given definite integral. $$ \int_{1}^{e} x \ln (x) d x $$

Evaluate the given definite integral. $$ \int_{0}^{\pi} \sin ^{3}(x) \cos ^{4}(x) d x $$

Decompose the given rational function into partial fractions. Calculate the coefficients. $$ \frac{2 x^{3}+x^{2}-5 x+2}{x^{2}(x+1)(x-2)} $$

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{0}^{3} x^{-1 / 2}(1+x) d x\)

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

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int \frac{1}{\left(1+x^{2}\right)^{3 / 2}} d x $$

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int \frac{1}{\left(5+x^{2}\right)^{3 / 2}} d x $$