Chapter 7

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

We can show that the Taylor polynomial for \(f(x)=\tan ^{-1} x\) about \(x=0\) converges for \(|x| \leq 1\). (a) Show that the following is true: $$ \tan ^{-1} x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\cdots+R_{n+1}(x) $$ (b) Explain why the following holds: $$ \frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots $$ (This series converges very slowly, as you would see if you used it to approximate \(\pi .\) )

Although we cannot compute the antiderivative of \(f(x)=e^{-x^{2} / 2}\), it can be shown that: $$\int_{-\infty}^{\infty} e^{-x^{2} / 2} d x=\sqrt{2 \pi}$$ Use this fact to show that $$\int_{-\infty}^{\infty} x^{2} e^{-x^{2} / 2} d x=\sqrt{2 \pi}$$ Hint: Write the integrand as \(x \cdot\left(x e^{-x^{2} / 2}\right)\) and use integration by parts.

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

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

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

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 trapezoidal rule approximation of $$ \int_{0}^{1} e^{-x} d x $$ is accurate to within \(10^{-5}\) ?

Determine the constant \(c\) so that $$ \int_{0}^{\infty} c e^{-3 x} d x=1 $$

Evaluating the integral \(\int \arcsin x d x\) requires two steps. (a) Write $$ \arcsin x=1 \cdot \arcsin x $$ and integrate by parts once to show that $$ \int \arcsin x d x=x \arcsin x-\int \frac{x}{\sqrt{1-x^{2}}} d x $$ (b) Use substitution to compute $$ \int \frac{x}{\sqrt{1-x^{2}}} d x $$ and combine your result in (a) with (7.11) to complete the computation of \(\int \arcsin x d x\).

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

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

Determine the constant \(c\) so that $$ \int_{-\infty}^{\infty} \frac{c}{1+x^{2}} d x=1 $$

Evaluating the integral \(\int \arccos x d x\) requires two steps. (a) Write $$ \arccos x=1 \cdot \arccos x $$ and integrate by parts once to show that $$ \int \arccos x d x=x \arccos x+\int \frac{x}{\sqrt{1-x^{2}}} d x $$ (b) Use substitution to compute $$ \int \frac{x}{\sqrt{1-x^{2}}} d x $$ and combine your result in (a) with (7.12) to complete the computation of \(\int \arccos x d x\).

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

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

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 trapezoidal rule approximation of $$ \int_{1}^{2} \frac{e^{t}}{t} d t $$ is accurate to within \(10^{-4}\) ?

In this problem, we investigate the integral \(\int_{1}^{\infty} \frac{1}{x^{p}} d x\) for \(01\), set \(A(z)=\int_{1}^{z} \frac{1}{x^{p}} d x\) and show that $$A(z)=\frac{1}{1-p}\left(z^{-p+1}-1\right) \quad \text { for } p \neq 1$$ and $$A(z)=\ln z \quad \text { for } p=1$$ (b) Use your results in (a) to show that, for \(0

1\), $$ \lim _{z \rightarrow \infty} A(z)=\frac{1}{p-1} $$

(a) Use integration by parts to show that, for \(x>0\), $$ \int \frac{1}{x} \ln x d x=(\ln x)^{2}-\int \frac{1}{x} \ln x d x $$ (b) Use your result in (a) to evaluate $$ \int \frac{1}{x} \ln x d x $$

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

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

In this problem, we investigate the integral \(\int_{0}^{1} \frac{1}{x p} d x\) for \(0

(a) Use integration by parts to show that $$ \int x^{n} e^{x} d x=x^{n} e^{x}-n \int x^{n-1} e^{x} d x $$ Such formulas are called reduction formulas, since they reduce the exponent of \(x\) by 1 each time they are applied. (b) Apply the reduction formula in (a) repeatedly to compute \(\int x^{3} e^{x} d x\)

Use partial-fraction decomposition to evaluate the integrals. $$ \int \frac{4 x^{2}-x-1}{(x+1)(x-3)} d x $$

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

(a) Show that $$0 \leq e^{-x^{2}} \leq e^{-x}$$ for \(x \geq 1\). (b) Use your result in (a) to show that $$\int_{1}^{\infty} e^{-x^{2}} d x$$ is convergent.

(a) Use integration by parts to verify the validity of the reduction formula $$ \int x^{n} e^{a x} d x=\frac{1}{a} x^{n} e^{e x}-\frac{n}{a} \int x^{n-1} e^{a x} d x $$ where \(a\) is a constant not equal to 0 . (b) Apply the reduction formula in (a) to compute \(\int x^{2} e^{-3 x} d x\).

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{2 x-3}{(x-1)^{2}} d x $$

In Problems \(37-42, a, b\), and \(c\) are constants and \(g(x)\) is a continuous function whose derivative \(g^{\prime}(x)\) is also continuous. Use substitution to evaluate each indefinite integral. $$ \int \frac{2 a x+b}{a x^{2}+b x+c} d x $$

Using a spreadsheet, approximate the integral \(\int_{0}^{1} \sqrt{x} d x\) using the midpoint rule and: (a) \(n=5\) subintervals, (b) \(n=10\) subintervals, (c) \(n=20\) subintervals, (d) \(n=50\) subintervals. (e) What is the exact value of the integral? (f) By comparing your answers from (a)-(d) with the exact answer, calculate the error \(L_{n}=\left|M_{n}-\int_{0}^{1} \sqrt{x} d x\right|\), and make a plot of \(L_{n}\) against \(n\) using your data. (g) By plotting \(\log L_{n}\) against \(\log n\), show how your data support the claim that the error decreases proportional to \(1 / n^{2}\).

(a) Show that $$0 \leq \frac{1}{\sqrt{1+x^{4}}} \leq \frac{1}{x^{2}}$$ for \(x>0\). (b) Use your result in (a) to show that $$\int_{1}^{\infty} \frac{1}{\sqrt{1+x^{4}}} d x$$ is convergent.

(a) Use integration by parts to verify the validity of the reduction formula $$ \int(\ln x)^{n} d x=x(\ln x)^{n}-n \int(\ln x)^{n-1} d x $$ (b) Apply the reduction formula in (a) repeatedly to compute $$ \int(\ln x)^{3} d x $$

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{x-1}{(x+1)^{2}} d x $$

In Problems \(37-42, a, b\), and \(c\) are constants and \(g(x)\) is a continuous function whose derivative \(g^{\prime}(x)\) is also continuous. Use substitution to evaluate each indefinite integral. $$ \int \frac{1}{a x+b} d x $$

(a) Show that $$\frac{1}{\sqrt{1+x^{2}}} \geq \frac{1}{2 x}>0$$ for \(x \geq 1\). (b) Use your result in (a) to show that $$\int_{1}^{\infty} \frac{1}{\sqrt{1+x^{2}}} d x$$ is divergent.

In Problems \(39-48\), first make an appropriate substitution and then use integration by parts to evaluate each integral. $$ \int e^{\sqrt{x}} d x $$

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{3 x^{2}-x+1}{x(x-1)^{2}} d x $$

In Problems \(37-42, a, b\), and \(c\) are constants and \(g(x)\) is a continuous function whose derivative \(g^{\prime}(x)\) is also continuous. Use substitution to evaluate each indefinite integral. $$ \int g^{\prime}(x)[g(x)]^{n} d x $$

(a) Show that $$0 \leq \frac{1}{\sqrt{x+x^{4}}} \leq \frac{1}{x^{2}}$$ for \(x>0\). (b) Use your result in (a) to show that $$\int_{1}^{\infty} \frac{1}{\sqrt{x+x^{4}}} d x$$ is convergent.

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{4 x^{2}+3 x+1}{(x+1)^{2}(x-1)} d x $$

In Problems \(37-42, a, b\), and \(c\) are constants and \(g(x)\) is a continuous function whose derivative \(g^{\prime}(x)\) is also continuous. Use substitution to evaluate each indefinite integral. $$ \int g^{\prime}(x) \cos [g(x)] d x $$

Find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{0}^{\infty} e^{-x^{2} / 2} d x $$

In Problems \(39-48\), first make an appropriate substitution and then use integration by parts to evaluate each integral. $$ \int x^{3} e^{-x^{2} / 2} d x $$

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{x^{2}-2 x-2}{x^{2}(x+2)} \cdot d x $$

In Problems \(37-42, a, b\), and \(c\) are constants and \(g(x)\) is a continuous function whose derivative \(g^{\prime}(x)\) is also continuous. Use substitution to evaluate each indefinite integral. $$ \int \frac{g^{\prime}(x)}{[g(x)]^{2}+1} d x $$

Find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{1}^{\infty} \frac{1}{\left(1+x^{4}\right)^{1 / 3}} d x $$

In Problems \(39-48\), first make an appropriate substitution and then use integration by parts to evaluate each integral. $$ \int x^{5} e^{x^{2}} d x $$

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{4 x^{2}-x-1}{(x+1)^{2}(x-3)} d x $$

In Problems \(37-42, a, b\), and \(c\) are constants and \(g(x)\) is a continuous function whose derivative \(g^{\prime}(x)\) is also continuous. Use substitution to evaluate each indefinite integral. $$ \int g^{\prime}(x) e^{-g(x)} d x $$

Find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{1}^{\infty} \frac{1}{\sqrt{1+x}} d x $$

In Problems \(39-48\), first make an appropriate substitution and then use integration by parts to evaluate each integral. $$ \int x^{3} \sin \left(x^{2}\right) d x $$