Chapter 7

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

In Problems \(6-10\), compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions. $$ f(x)=x^{5}, n=6 $$

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

In Problems 9-12, write out the partial-fraction decomposition of the function \(f(x)\). $$ f(x)=\frac{5 x-1}{x^{2}-1} $$

Use integration by parts to evaluate the integrals. $$ \int 2 x^{2} e^{-x} d x $$

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

Use the trapezoidal rule to approximate each integral with the specified value of \(n .\) \(\int_{-1}^{0} x^{3} d x, n=5\)

In Problems \(6-10\), compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions. $$ f(x)=\sqrt{1+x}, n=3 $$

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

In Problems , write out the partial-fraction decomposition of the function \(f(x)\). $$ f(x)=\frac{9 x-7}{2 x^{2}-7 x+3} $$

Use integration by parts to evaluate the integrals. $$ \int x \ln x d x $$

Evaluate the indefinite integral by making the given substitution. $$ \int x e^{-x^{2} / 2} d x, \text { with } u=-x^{2} / 2 $$

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

In Problems 11-16, compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions and compare the value of the functions at the indicated point with the value of the corresponding Taylor polynomial. $$ f(x)=\sqrt{2+x}, n=3, x=0.1 $$

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

In Problems , write out the partial-fraction decomposition of the function \(f(x)\). $$ f(x)=\frac{4 x+1}{x^{2}-3 x-10} $$

Use integration by parts to evaluate the integrals. $$ \int x^{2} \ln x d x $$

Evaluate the indefinite integral by making the given substitution. $$ \int x e^{1-3 x^{2}} d x, \text { with } u=1-3 x^{2} $$

Use the trapezoidal rule to approximate each integral with the specified value of \(n .\) \(\int_{0}^{\pi / 2} \sin x d x, n=4\)

In Problems 11-16, compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions and compare the value of the functions at the indicated point with the value of the corresponding Taylor polynomial. $$ f(x)=\frac{1}{1-x}, n=3, x=0.1 $$

In Problems , write out the partial-fraction decomposition of the function \(f(x)\).$$ f(x)=-\frac{10}{3 x^{2}+8 x-3} $$

Use integration by parts to evaluate the integrals. $$ \int x \ln (3 x) d x $$

Evaluate the 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 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 11-16, compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions and compare the value of the functions at the indicated point with the value of the corresponding Taylor polynomial. $$ f(x)=\sin x, n=5, x=1 $$

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{0}^{\pi / 2} \frac{\cos x}{\sqrt{\sin x}} d x $$

In Problems 13-18, use partial-fraction decomposition to evaluate the integrals. $$ \int \frac{1}{x(x-2)} d x $$

Use integration by parts to evaluate the integrals. $$ \int x^{2} \ln x^{2} d x $$

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

Evaluate the indefinite integral by making the given substitution. $$ \int \frac{2 x}{3-x^{2}} d x, \text { with } u=3-x^{2} $$

In Problems 11-16, compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions and compare the value of the functions at the indicated point with the value of the corresponding Taylor polynomial. $$ f(x)=e^{-x}, n=4, x=0.3 $$

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{-2}^{0} \frac{d x}{(x+1)^{1 / 3}} $$

In Problems , use partial-fraction decomposition to evaluate the integrals. $$ \int \frac{1}{x(2 x+1)} d x $$

Use integration by parts to evaluate the integrals. $$ \int x \sec ^{2} x d x $$

Evaluate the indefinite integral by making the given substitution. $$ \int \frac{3 x}{x+4} d x, \text { with } u=x+4 $$

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

In Problems 11-16, compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions and compare the value of the functions at the indicated point with the value of the corresponding Taylor polynomial. $$ f(x)=\tan x, n=2, x=0.1 $$

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

In Problems , use partial-fraction decomposition to evaluate the integrals. $$ \int \frac{1}{(x+1)(x-3)} d x $$

Use integration by parts to evaluate the integrals. $$ \int x \csc ^{2} x d x $$

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

Evaluate the indefinite integral by making the given substitution. $$ \int \frac{x}{5-x} d x, \text { with } u=5-x $$

In Problems 11-16, compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions and compare the value of the functions at the indicated point with the value of the corresponding Taylor polynomial. $$ f(x)=\ln (1+x), n=3, x=0.1 $$

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

In Problems , use partial-fraction decomposition to evaluate the integrals.$$ \int \frac{1}{(x-1)(x+2)} d x $$

Use integration by parts to evaluate the integrals. $$ \int_{0}^{\pi / 3} x \sin x d x $$

Use substitution to evaluate the indefinite integrals. $$ \int \sqrt{x+3} d x $$

How large should \(n\) be so that the midpoint rule approximation of $$ \int_{0}^{2} x^{2} d x $$ is accurate to within \(10^{-4} ?\) In Problems 18-24, use the theoretical error bound to determine how large \(n\) should be [Hint: In each case, find the second derivative of the integrand, graph it, and use a graphing calculator to find an upper bound on \(\left.\left|f^{\prime \prime}(x)\right| .\right]\)

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

In Problems \(17-28\), determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{1}^{\infty} \frac{1}{x^{3}} d x $$