91Ó°ÊÓ

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{0}^{-9} \frac{d x}{\sqrt{9-x}} $$

Evaluate the indefinite integral by making the given substitution. $$ \int x e^{-x^{2} / 2} d x, \text { with } u=-x^{2} / 2 $$

In Problems , write out the partial-fraction decomposition of the function \(f(x)\). $$ f(x)=\frac{4 x+1}{x^{2}-3 x-10} $$

Use the trapezoidal rule to approximate each integral with the specified value of \(n .\) \(\int_{0}^{1} e^{-x} d x, n=3\)

Use integration by parts to evaluate the integrals. $$ \int x \ln x d x $$

In Problems 11-16, compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions and compare the value of the functions at the indicated point with the value of the corresponding Taylor polynomial. $$ f(x)=\sqrt{2+x}, n=3, x=0.1 $$

Use integration by parts to evaluate the integrals. $$ \int x^{2} \ln x d x $$

In Problems 11-16, compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions and compare the value of the functions at the indicated point with the value of the corresponding Taylor polynomial. $$ f(x)=\frac{1}{1-x}, n=3, x=0.1 $$

Evaluate the indefinite integral by making the given substitution. $$ \int x e^{1-3 x^{2}} d x, \text { with } u=1-3 x^{2} $$

Use the trapezoidal rule to approximate each integral with the specified value of \(n .\) \(\int_{0}^{\pi / 2} \sin x d x, n=4\)