91Ó°ÊÓ

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. $$ \int_{-2}^{0} \frac{d x}{(x+1)^{1 / 3}} $$

Use the trapezoidal rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. \(\int_{-1}^{1}\left(1-e^{-x}\right) d x, n=4\)

Use the trapezoidal rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. \(\int_{0}^{2} \sqrt{x} d x, n=4\)

Evaluate the indefinite integral by making the given substitution. $$ \int \frac{3 x}{x+4} d x, \text { with } u=x+4 $$

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

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. $$ f(x)=\tan x, n=2, x=0.1 $$

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{-1}^{1} \ln |x| d x $$

In Problems , use partial-fraction decomposition to evaluate the integrals. $$ \int \frac{1}{(x+1)(x-3)} d x $$

Evaluate the indefinite integral by making the given substitution. $$ \int \frac{x}{5-x} d x, \text { with } u=5-x $$