Chapter 7

$$ \text { In Problems } 27-36, \text { evaluate each integral. } $$ $$ \int \frac{1}{(x-3)(x+2)} d x $$

Use substitution to evaluate the indefinite integrals. $$ \int \sec ^{2} x e^{\tan x} d x $$

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

$$ \text { In Problems } , \text { evaluate each integral. } $$ $$ \int \frac{2 x-1}{(x+4)(x+1)} d x $$

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

Use substitution to evaluate the indefinite integrals. $$ \int \sin \left(\frac{3 \pi}{2} x+\frac{\pi}{4}\right) d x $$

In Problems 27-30, use the following form of the error term $$ R_{n+1}(x)=\frac{f^{(n+1)}(c)}{(n+1) !} x^{n+1} $$ where \(c\) is between 0 and \(x\), to determine in advance the degree of Taylor polynomial at \(a=0\) that would achieve the indicated accuracy in the interval \([0, x] .\) (Do not compute the Taylor polynomial.) $$ f(x)=1 /(1+x), x=0.2, \text { error }<10^{-2} $$

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

$$ \text { In Problems } , \text { evaluate each integral. } $$ $$ \int \frac{1}{x^{2}-9} d x $$

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

Use substitution to evaluate the indefinite integrals. $$ \int \cos (2 x-1) d x $$

In Problems 27-30, use the following form of the error term $$ R_{n+1}(x)=\frac{f^{(n+1)}(c)}{(n+1) !} x^{n+1} $$ where \(c\) is between 0 and \(x\), to determine in advance the degree of Taylor polynomial at \(a=0\) that would achieve the indicated accuracy in the interval \([0, x] .\) (Do not compute the Taylor polynomial.) $$ f(x)=\ln (1+x), x=0.1, \text { error }<10^{-2} $$

Although we cannot compute the antiderivative of \(f(x)=\) \(e^{-x^{2} / 2}\), it is known 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.

$$ \text { In Problems } , \text { evaluate each integral. } $$ $$ \int \frac{1}{x^{2}+9} d x $$

Evaluating the integral $$ \int \cos ^{2} x d x $$ 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 substitution to evaluate the indefinite integrals. $$ \int \tan x \sec ^{2} x d x $$

Let \(f(x)=e^{-1 / x}\) for \(x>0\) and \(f(x)=0\) for \(x=0\). Compute a Taylor polynomial of degree 2 at \(x=0\), and determine how large the error is.

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

$$ \text { In Problems } , \text { evaluate each integral. } $$ $$ \int \frac{1}{x^{2}-x-2} d x $$

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 substitution to evaluate the indefinite integrals. $$ \int \sin ^{3} x \cos x d x $$

We can show that the Taylor polynomial for \(f(x)=(1+x)^{\alpha}\) about \(x=0\), with \(\alpha\) a positive constant, converges for \(x \in(-1,1)\). Show that $$ \begin{aligned} (1+x)^{\alpha}=& 1+\alpha x+\frac{\alpha(\alpha-1)}{2 !} x^{2} \\ &+\frac{\alpha(\alpha-1)(\alpha-2)}{3 !} x^{3}+\cdots+R_{n+1}(x) \end{aligned} $$

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

$$ \text { In Problems } , \text { evaluate each integral. } $$ $$ \int \frac{1}{x^{2}-x+2} d x $$

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 (7.7) to complete the computation of \(\int \arcsin x d x\).

Use substitution to evaluate the indefinite integrals. $$ \int \frac{(\ln x)^{2}}{x} d x $$

In this problem, we investigate the integral $$ \int_{1}^{\infty} \frac{1}{x^{p}} d x $$

$$ \text { In Problems } , \text { evaluate each integral. } $$ $$ \int \frac{x^{2}+1}{x^{2}+3 x+2} d x $$

Use substitution to evaluate the indefinite integrals. $$ \int \frac{d x}{(x-3) \ln (x-3)} $$

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

$$ \text { In Problems } , \text { evaluate each integral. } $$ $$ \int \frac{x^{3}+1}{x^{2}+3} d x $$

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 substitution to evaluate the indefinite integrals. $$ \int x^{3} \sqrt{5+x^{2}} d x $$

(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.

$$ \text { In Problems } , \text { evaluate each integral. } $$ $$ \int \frac{x^{2}+4}{x^{2}-4} d x $$

Use substitution to evaluate the indefinite integrals. $$ \int \sqrt{1+\ln x} \frac{\ln x}{x} d x $$

(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.

$$ \text { In Problems } , \text { evaluate each integral. } $$ $$ \int \frac{x^{4}+3}{x^{2}-4 x+3} d x $$

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

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 the indefinite integrals. $$ \int \frac{2 a x+b}{a x^{2}+b x+c} d x $$

(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.

$$ \text { In Problems } 37-44, \text { evaluate each definite integral. } $$ $$ \int_{3}^{5} \frac{x-1}{x} d x $$

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 $$ \int(\ln x)^{3} d x $$

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 the indefinite integrals. $$ \int \frac{1}{a x+b} d x $$

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

$$ \text { In Problems } , \text { evaluate each definite integral. } $$ $$ \int_{3}^{5} \frac{x}{x-1} d x $$

First make an appropriate substitution and then use integration by parts to evaluate the indefinite integrals. $$ \int \cos \sqrt{x} d x $$

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 the indefinite integrals. $$ \int g^{\prime}(x)[g(x)]^{n} d x $$

In Problems 39-42, find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{-\infty}^{\infty} e^{-x^{2} / 2} d x $$

$$ \text { In Problems } , \text { evaluate each definite integral. } $$ $$ \int_{0}^{1} \frac{x}{x^{2}+1} d x $$