91Ó°ÊÓ

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. $$ f(x)=\frac{1}{1-x}, n=3, x=0.1 $$

Use the trapezoidal rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. \(\int_{1}^{3} x^{3} d x, n=5\)

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{0}^{\pi / 2} \frac{\cos x}{\sqrt{\sin x}} d x $$

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. $$ f(x)=\sin x, n=5, x=1 $$

In Problems 13-18, use partial-fraction decomposition to evaluate the integrals. $$ \int \frac{1}{x(x-2)} d x $$

Use integration by parts to evaluate the integrals. $$ \int x \ln (3 x) d x $$

Evaluate the indefinite integral by making the given substitution. $$ \int \frac{x+2}{x^{2}+4 x} d x, \text { with } u=x^{2}+4 x $$

In Problems , use partial-fraction decomposition to evaluate the integrals. $$ \int \frac{1}{x(2 x+1)} d x $$

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

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{-2}^{0} \frac{d x}{(x+1)^{1 / 3}} $$