Chapter 7

Use the midpoint rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. $$ \int_{1}^{3} \frac{2}{\sqrt{x}} d x, n=5 $$

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

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int 3 x e^{-2 x} d x $$

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{3}+2 x+3}{x+1} $$

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

Use the Table of Integrals to compute each integral after manipulating the integrand in a suitable way. $$ \int_{0}^{\pi / 2} e^{x} \cos \left(x-\frac{\pi}{6}\right) d x $$

Compute the Taylor polynomial of degree \(n\) about \(x=0\) for each function. $$ f(x)=x^{5}, n=6 $$

Use the trapezoidal rule to approximate each integral with the specified value of \(n\). $$ \int_{1}^{2} \sqrt{x^{2}+1} d x, n=4 $$

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

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

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{3}+3 x^{2}+3 x+1}{x^{2}+1} $$

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

Use the Table of Integrals to compute each integral after manipulating the integrand in a suitable way. $$ \int_{1}^{2} x \ln (x+3) d x $$

Compute the Taylor polynomial of degree \(n\) about \(x=0\) for each function. $$ f(x)=\sqrt{1+x}, n=3 $$

Use the trapezoidal rule to approximate each integral with the specified value of \(n\). $$ \int_{-1}^{0} \sin \left(x^{2}\right) d x, n=5 $$

All the integrals in problem are improper and converge. Explain in each case why the integral is improper, and evaluate each integral. $$ \int_{-\infty}^{\infty} x^{3} e^{-x^{4}} d x $$

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

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{3}+1}{x^{2}+x+1} $$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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. $$ \int \frac{x^{2}}{x^{2}+4 x+1} d x $$

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. $$ f(x)=e^{2 x}, n=4, x=0.3 $$

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

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

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

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

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

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

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