91Ó°ÊÓ

Use the Table of Integrals to compute each integral after manipulating the integrand in a suitable way. $$ \int \frac{x^{2}}{x^{2}+4 x+1} d x $$

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{5}-1}{x-1} $$

Compute the Taylor polynomial of degree \(n\) about \(x=0\) for each function and compare the value of the function at the indicated point with the value of the corresponding Taylor polynomial. $$ f(x)=e^{2 x}, n=4, x=0.3 $$

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

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int \frac{3 x}{1-x^{2}} d x, \text { with } u=1-x^{2} $$

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

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int \frac{3 x}{x+2} d x, \text { with } u=x+2 $$

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int x \sec ^{2} x d x $$

Use the Table of Integrals to compute each integral after manipulating the integrand in a suitable way. $$ \int \sqrt{x^{2}+2 x+2} d x $$

Compute the Taylor polynomial of degree \(n\) about \(x=0\) for each function and compare the value of the function at the indicated point with the value of the corresponding Taylor polynomial. $$ f(x)=\tan x, n=2, x=0.1 $$