Chapter 7

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

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

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

Use the Table of Integrals to compute each integral after manipulating the integrand in a suitable way. $$ \int \frac{1}{\sqrt{16-9 x^{2}}} d x $$

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. $$ f(x)=\ln (1+x), n=3, x=0.1 $$

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

All the integrals in problem are improper and converge. Explain in each case why the integral is improper, and evaluate each integral. $$ \int_{0}^{2} \frac{d x}{(x-1)^{2 / 5}} $$

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

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

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

(a) Find the Taylor polynomial of degree 3 about \(x=0\) for \(f(x)=\sin x\). (b) Use your result in (a) to explain why $$ \lim _{x \rightarrow 0} \frac{\sin x}{x}=1 $$

Use a spreadsheet to approximate each of the following integrals using the midpoint rule with each of the specified values of \(n .\) \(\int_{0}^{1} x^{2} d x\) (a) \(n=10\) (b) \(n=20\)

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{0}^{\infty} \frac{1}{x+1} d x $$

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int_{0}^{\pi / 3} x \sin x d x $$

Write out the partial-fraction decomposition of the function \(f(x)\). $$ f(x)=\frac{2 x-3}{x(x+1)} $$

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int \sqrt{x+2} d x $$

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

(a) Find the Taylor polynomial of degree 2 about \(x=0\) for \(f(x)=\cos x\) (b) Use your result in (a) to explain why $$ \lim _{x \rightarrow 0} \frac{\cos x-1}{x}=0 $$

Use a spreadsheet to approximate each of the following integrals using the midpoint rule with each of the specified values of \(n .\) \(\int_{-1}^{1} \sqrt{x+1} d x\) (a) \(n=10\) (b) \(n=30\).

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{-1}^{0} \frac{1}{x+1} d x $$

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int_{0}^{\pi / 4} x \cos 2 x d x $$

Write out the partial-fraction decomposition of the function \(f(x)\). $$ f(x)=-\frac{x+1}{(2 x+1)(x-1)} $$

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int(4+x)^{1 / 7} d x $$

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

Compute the Taylor polynomial of degree \(n\) about a and compare the value of the approximation with the value of the function at the given point \(x\). $$ f(x)=\sqrt{x}, a=1, n=3 ; x=2 $$

Use a spreadsheet to approximate each of the following integrals using the midpoint rule with each of the specified values of \(n .\) \(\int_{0}^{1} \sin \left(x^{2}\right) d x\) (a) \(n=10\) (b) \(n=20\).

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{1}^{\infty} \frac{1}{x^{3}} d x $$

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

Write out the partial-fraction decomposition of the function \(f(x)\). $$ f(x)=\frac{x+1}{x(x+2)} $$

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int(4 x-1) \sqrt{2 x^{2}-x+2} d x $$

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

Compute the Taylor polynomial of degree \(n\) about a and compare the value of the approximation with the value of the function at the given point \(x\). $$ f(x)=\ln x, a=1, n=3 ; x=2 $$

Use a spreadsheet to approximate each of the following integrals using the midpoint rule with each of the specified values of \(n .\) \(\int_{0}^{1} e^{-\sqrt{x}} d x\) (a) \(n=10\) (b) \(n=40\).

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{1}^{\infty} \frac{1}{x^{1 / 3}} d x $$

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

Write out the partial-fraction decomposition of the function \(f(x)\). $$ f(x)=\frac{2 x+1}{(x+1)(x-1)} $$

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int\left(x^{2}-2 x\right)\left(x^{3}-3 x^{2}+2\right)^{2 / 3} d x $$

Compute the Taylor polynomial of degree \(n\) about a and compare the value of the approximation with the value of the function at the given point \(x\). $$ f(x)=\cos x, a=\frac{\pi}{2}, n=3 ; x=\frac{\pi}{3} $$

Use a spreadsheet to approximate each of the following integrals using the midpoint rule with each of the specified values of \(n .\) \(\int_{0}^{\pi} e^{-x} \cos x d x\) (a) \(n=20\) (b) \(n=40\).

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{0}^{4} \frac{1}{x^{4}} d x $$

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int_{1}^{4} \ln \sqrt{x} d x $$

Write out the partial-fraction decomposition of the function \(f(x)\). $$ f(x)=\frac{16 x-6}{(2 x-5)(3 x+1)} $$

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int \frac{2 x-2}{1+4 x-2 x^{2}} d x $$

Use the Table of Integrals to compute each integral after manipulating the integrand in a suitable way. $$ \int \frac{3}{x^{2}-4 x+5} d x $$

Compute the Taylor polynomial of degree \(n\) about a and compare the value of the approximation with the value of the function at the given point \(x\). $$ f(x)=x^{1 / 5}, a=1, n=3 ; x=0.9 $$

Use a spreadsheet to approximate each of the following integrals using the midpoint rule with each of the specified values of \(n .\) \(\int_{2}^{4} \frac{1}{\ln x} d x\) (a) \(n=20\) (b) \(n=50\).

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{0}^{4} \frac{1}{x^{1 / 4}} d x $$

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int_{1}^{4} \sqrt{x} \ln x d x $$

Write out the partial-fraction decomposition of the function \(f(x)\). $$ f(x)=\frac{4 x^{2}-14 x-6}{x(x-3)(x+1)} $$

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int \frac{x^{2}-1}{x^{3}-3 x+1} d x $$