91Ó°ÊÓ

Evaluate the given integral. $$ \int \sin ^{2}(x / 2) d x $$

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int \frac{x}{\sqrt{9-x^{2}}} d x $$

In each of Exercises \(1-10,\) determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{1}^{5}(x-5)^{-4 / 3} d x\)

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

In each of Exercises \(1-20\), determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{3}^{\infty} x^{-3 / 2} d x $$

In each of Exercises \(1-10,\) write down the form of the partial fraction decomposition of the given rational function. Do not explicitly calculate the coefficients. \(\frac{2 x^{3}+x+1}{\left(x^{2}+1\right)\left(x^{2}+4\right)}\)

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int \frac{1}{\sqrt{9-x^{2}}} d x $$

Write down the form of the partial fraction decomposition of the given rational function. Do not explicitly calculate the coefficients. \(\frac{2 x^{4}}{\left(x^{2}+2 x+2\right)\left(2 x^{2}+5 x+3\right)}\)

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{9}^{\infty} \frac{1}{\sqrt{x}} d x $$

Evaluate the given integral. $$ \int \sin ^{2}(4 x) d x $$