91Ó°ÊÓ

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int x \ln x d x $$

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. $$ f(x)=\sqrt{1+x}, n=3, x=0.1 $$

Use the Table of Integrals to compute each integral after manipulating the integrand in a suitable way. $$ \int\left(x^{2}-1\right) e^{-x / 2} d x $$

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{3}}{x^{2}+x} $$

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

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

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

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

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{3}+x}{x^{2}+x} $$

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