91Ó°ÊÓ

Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. $$ \int \frac{d x}{x^{2} \sqrt{4-x^{2}}} \quad x=2 \sin \theta $$

Evaluate the integral. $$ \int \sin ^{2} x \cos ^{3} x d x $$

Evaluate the integral. $$ \int \frac{\cos x}{1-\sin x} d x $$

Explain why each of the following integrals is improper. $$\begin{array}{ll}{\text { (a) } \int_{1}^{2} \frac{x}{x-1} d x} & {\text { (b) } \int_{0}^{\infty} \frac{1}{1+x^{3}} d x} \\ {\text { (c) } \int_{-\infty}^{\infty} x^{2} e^{-x^{2}} d x} & {\text { (d) } \int_{0}^{\pi / 4} \cot x d x}\end{array}$$

Evaluate the integral using integration by parts with the indicated choices of \(u\) and \(d v\) $$ \int x e^{2 x} d x ; u=x, d v=e^{2 x} d x $$

Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coeficients. \( \text { (a) } \frac{x-6}{x^{2}+x-6} \) \( \text { (b) } \frac{x^{2}}{x^{2}+x+6} \)

Evaluate the integral using integration by parts with the indicated choices of \(u\) and \(d v\) $$ \int \sqrt{x} \ln x d x ; \quad u=\ln x, \quad d v=\sqrt{x} d x $$

Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. $$ \int \frac{x^{3}}{\sqrt{x^{2}+4}} d x \quad x=2 \tan \theta $$

Evaluate the integral. $$ \int \sin ^{3} \theta \cos ^{4} \theta d \theta $$

Evaluate the integral. $$ \int_{0}^{1}(3 x+1)^{\sqrt{2}} d x $$