Chapter 7

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 \(x\). $$ f(x)=e^{x}, a=2, n=3 ; x=2.1 $$

Use a spreadsheet to approximate each of the following integrals using the trapezoidal rule with each of the specified values of \(n\). \(\int_{1}^{4} x^{3} d x\) (a) \(n=10\) (b) \(n=20\).

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{0}^{2} \frac{1}{(x-1)^{1 / 3}} d x $$

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

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

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

Show that $$ x^{4} \approx a^{4}+4 a^{3}(x-a) $$ for \(x\) close to \(a\).

Use a spreadsheet to approximate each of the following integrals using the trapezoidal rule with each of the specified values of \(n\). \(\int_{0}^{1}\left(x^{2}+1\right)^{1 / 3} d x\) (a) \(n=10\) (b) \(n=30\).

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

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int_{0}^{3} x^{2} e^{-x} d x $$

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

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int \frac{x^{3}-1}{x^{4}-4 x} d x $$

Show that, for positive constants \(r\) and \(K\), $$ r N\left(1-\frac{N}{K}\right) \approx r N $$ for \(N\) close to 0 .

Use a spreadsheet to approximate each of the following integrals using the \(\int_{0}^{\pi} x \sin x d x\) (a) \(n=10\) (b) \(n=20\).

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{0}^{2} \frac{1}{(x-1)^{4}} d x $$

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

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

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

(a) Show that, for positive constants \(a\) and \(k\), $$ f(R)=\frac{a R}{k+R} \approx \frac{a}{k} R $$ for \(R\) close to 0 . (b) Show that, for positive constants \(a\) and \(k\), $$ f(R)=\frac{a R}{k+R} \approx \frac{a}{2}+\frac{a}{4 k}(R-k) $$ for \(R\) close to \(k\).

Use a spreadsheet to approximate each of the following integrals using the trapezoidal rule with each of the specified values of \(n\). \(\int_{1}^{\pi} \frac{\cos x}{x} d x\) (a) \(n=15\) (b) \(n=30\).

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

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

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

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int \cos x e^{-\sin x} d x $$

Use the following form of the error term $$ \left|R_{n+1}(x)\right| \leq \frac{K|x|^{n+1}}{(n+1) !} $$ where \(K=\) largest value of \(\left|f^{(n+1)}(t)\right|\) for \(0 \leq t \leq x\), to determine in advance the degree of Taylor polynomial at \(a=0\) that would achieve the indicated accuracy in the interval \([\mathbf{0}, \boldsymbol{x}] .\) (Do not compute the Taylor polynomial.) $$ f(x)=e^{x}, x=2, \text { error }<10^{-3} $$

Use a spreadsheet to approximate each of the following integrals using the trapezoidal rule with each of the specified values of \(n\). \(\int_{1}^{5} \frac{e^{-x}}{x} d x\) (a) \(n=20\) (b) \(n=40\).

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

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

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

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

Use the following form of the error term $$ \left|R_{n+1}(x)\right| \leq \frac{K|x|^{n+1}}{(n+1) !} $$ where \(K=\) largest value of \(\left|f^{(n+1)}(t)\right|\) for \(0 \leq t \leq x\), to determine in advance the degree of Taylor polynomial at \(a=0\) that would achieve the indicated accuracy in the interval \([\mathbf{0}, \boldsymbol{x}] .\) (Do not compute the Taylor polynomial.) $$ f(x)=\cos x, x=1, \text { error }<10^{-2} $$

Use a spreadsheet to approximate each of the following integrals using the trapezoidal rule with each of the specified values of \(n\). \(\int_{0}^{1} \exp (\cos x) d x\) (a) \(n=20\) (b) \(n=50\).

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

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

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int \sec ^{2} x e^{\tan x} d x $$

Use the following form of the error term $$ \left|R_{n+1}(x)\right| \leq \frac{K|x|^{n+1}}{(n+1) !} $$ where \(K=\) largest value of \(\left|f^{(n+1)}(t)\right|\) for \(0 \leq t \leq x\), to determine in advance the degree of Taylor polynomial at \(a=0\) that would achieve the indicated accuracy in the interval \([\mathbf{0}, \boldsymbol{x}] .\) (Do not compute the Taylor polynomial.) $$ f(x)=1 /(1+x), x=0.2, \text { error }<10^{-2} $$

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

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{-2}^{2} \frac{2 x d x}{\left(x^{2}-1\right)^{1 / 3}} $$

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

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

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int \sin x \cos x d x $$

Use the following form of the error term $$ \left|R_{n+1}(x)\right| \leq \frac{K|x|^{n+1}}{(n+1) !} $$ where \(K=\) largest value of \(\left|f^{(n+1)}(t)\right|\) for \(0 \leq t \leq x\), to determine in advance the degree of Taylor polynomial at \(a=0\) that would achieve the indicated accuracy in the interval \([\mathbf{0}, \boldsymbol{x}] .\) (Do not compute the Taylor polynomial.) $$ f(x)=\ln (1+x), x=0.1, \text { error }<10^{-2} $$

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{-\infty}^{1} \frac{3}{1+x^{2}} d x $$

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

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

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int \cos (2 x-1) d x $$

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

Determine whether $$\int_{-\infty}^{\infty} \frac{1}{x^{2}-1} d x$$ is convergent. Hint: Use the partial-fraction decomposition $$ \frac{1}{x^{2}-1}=\frac{1}{2}\left(\frac{1}{x-1}-\frac{1}{x+1}\right) $$

Evaluating the integral \(\int \cos ^{2} x d x\) requires two steps. First, write $$ \cos ^{2} x=(\cos x)(\cos x) $$ and integrate by parts to show that $$ \int \cos ^{2} x d x=\sin x \cos x+\int \sin ^{2} x d x $$ Then, use \(\sin ^{2} x+\cos ^{2} x=1\) to replace \(\sin ^{2} x\) in the integral on the right-hand side, and complete the integration of \(\int \cos ^{2} x d x\).

Use partial-fraction decomposition to evaluate the integrals. $$ \int \frac{1}{x(x-2)} d x $$