Chapter 7

Use the Table of Integrals to compute each integral. $$ \int \frac{x}{2 x+3} d x $$

Find the linear approximation of \(f(x)\) at \(x=0 .\) $$ f(x)=e^{x+1} $$

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

All the integrals in problem are improper and converge. Explain in each case why the integral is improper, and evaluate each integral. $$ \int_{0}^{\infty} 3 e^{-x} d x $$

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

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

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

Use the Table of Integrals to compute each integral. $$ \int \frac{d x}{4+x^{2}} $$

Find the linear approximation of \(f(x)\) at \(x=0 .\) $$ f(x)=\sin (x+1) $$

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

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

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

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

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int 4 x^{3} \sqrt{x^{4}+1} d x, \text { with } u=x^{4}+1 $$

Use the Table of Integrals to compute each integral. $$ \int \sqrt{x^{2}+16} d x $$

Find the linear approximation of \(f(x)\) at \(x=0 .\) $$ f(x)=\frac{1}{1+x} $$

Use the midpoint rule to approximate each integral with the specified value of \(n\). $$ \int_{0}^{1} \exp \left(x^{2}\right) d x, n=3 $$

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

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

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{2 x^{2}+5 x-1}{x+2} $$

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int 3 x\left(1+x^{2}\right)^{1 / 4} d x, \text { with } u=1+x^{2} $$

Use the Table of Integrals to compute each integral. $$ \int \sin (2 x) \cos (2 x) d x $$

Find the linear approximation of \(f(x)\) at \(x=0 .\) $$ f(x)=x^{4} $$

Use the midpoint rule to approximate each integral with the specified value of \(n\). $$ \int_{0}^{1} \sin \left(x^{2}\right) d x, n=4 $$

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

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=-\frac{x^{2}-4 x-1}{x-1} $$

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int 4 x^{3}\left(4-x^{4}\right)^{1 / 3} d x, \text { with } u=4-x^{4} $$

Use the Table of Integrals to compute each integral. $$ \int_{0}^{1} x^{3} e^{2 x} d x $$

Find the linear approximation of \(f(x)\) at \(x=0 .\) $$ f(x)=\tan x $$

Use the midpoint rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. $$ \int_{1}^{2} \frac{1}{x} d x, n=4 $$

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

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int 2 x \sin \left(\frac{x}{2}\right) d x $$

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{3 x^{3}+5 x-2 x^{2}-2}{x^{2}+1} $$

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int 5 \sin (2 x) d x, \text { with } u=2 x $$

Use the Table of Integrals to compute each integral. $$ \int_{0}^{x / 2} e^{-x} \cos (x) d x $$

Compute the Taylor polynomial of degree \(n\) about \(x=0\) for each function. $$ f(x)=\frac{1}{1+x}, n=4 $$

Use the midpoint rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. $$ \int_{0}^{1}\left(e^{2 x}-1\right) d x, n=4 $$

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

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int x \sin (1-2 x) d x $$

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{3}-3 x^{2}-15}{x^{2}+x+3} $$

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int 5 \sin (1-2 x) d x, \text { with } u=1-2 x $$

Use the Table of Integrals to compute each integral. $$ \int_{1}^{e} x^{2} \ln x d x $$

Compute the Taylor polynomial of degree \(n\) about \(x=0\) for each function. $$ f(x)=(1+x)^{3}, n=5 $$

Use the midpoint rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. $$ \int_{0}^{4} x^{3} d x, n=4 $$

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

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

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

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int 7 x \sin \left(4 x^{2}\right) d x \text { , with } u=4 x^{2} $$

Use the Table of Integrals to compute each integral. $$ \int_{e}^{e^{2}} \frac{\ln x}{x} d x $$

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