91Ó°ÊÓ

I-6 Write out the form of the partial fraction decomposition of the function (as in Example \(7 ) .\) Do not determine the numerical values of the coefficients. (a) $$\frac{2 x}{(x+3)(3 x+1)} \quad\( (b) \)\frac{1}{x^{3}+2 x^{2}+x}$$

Explain why each of the following integrals is improper. (a) $$ \int_{1}^{\infty} x^{4} e^{-x^{2}} d x $$ (b) $$\int_{0}^{\pi / 2} \sec x d x$$ (c) $$ \int_{0}^{2} \frac{x}{x^{2}-5 x+6} d x $$ (d) $$ \int_{-\infty}^{0} \frac{1}{x^{2}+5} d x $$

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

\(1-3\) Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. $$\int \frac{1}{x^{2} \sqrt{x^{2}-9}} d x ; \quad x=3 \sec \theta$$

1-2 Evaluate the integral using integration by parts with the indicated choices of \(u\) and \(d v .\) $$\int x^{2} \ln x d x ; \quad u=\ln x, d v=x^{2} d x$$

\(1-80\) Evaluate the integral. $$\int \cos x\left(1+\sin ^{2} x\right) d x$$

\(1-3\) Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. $$\int x^{3} \sqrt{9-x^{2}} d x ; \quad x=3 \sin \theta$ta$$

Which of the following integrals are improper? Why? $$\begin{array}{ll}{\text { (a) } \int_{1}^{2} \frac{1}{2 x-1} d x} & {\text { (b) } \int_{0}^{1} \frac{1}{2 x-1} d x} \\ {\text { (c) } \int_{-\infty}^{\infty} \frac{\sin x}{1+x^{2}} d x} & {\text { (d) } \int_{1}^{2} \ln (x-1) d x}\end{array}$$

\(1-80\) Evaluate the integral. $$\int \frac{\sin ^{3} x}{\cos x} d x$$

Evaluate the integral. $$\int \sin ^{6} x \cos ^{3} x d x$$