91Ó°ÊÓ

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

Use the trapezoidal rule to approximate each integral with the specified value of \(n\). $$ \int_{0}^{1} \sin (\sqrt{x}) d x, n=4 $$

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

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int x \ln (3 x) d x $$

All the integrals in problem are improper and converge. Explain in each case why the integral is improper, and evaluate each integral. $$ \int_{-2}^{2} \frac{d x}{\sqrt{4-x^{2}}} $$

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{4}+1}{x-1} $$

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

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int \frac{x+2}{x^{2}+4 x} d x, \text { with } u=x^{2}+4 x $$

Use the Table of Integrals to compute each integral after manipulating the integrand in a suitable way. $$ \int \cos ^{2}(5 x-3) d x $$

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