Techniques of Integration

7.5 Assess Your Understanding
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Concepts and Vocabulary

Multiple Choice A rational function $R(x)=\frac{p(x)}{q(x)}$ is proper when the degree of $p$ is [(a) less than, (b) equal to, (c) greater than] the degree of $q.$

(a)

True or False Every improper rational function can be written as the sum of a polynomial and a proper rational function.

True

True or False Sometimes the integration of a proper rational function leads to a logarithm.

True

True or False The decomposition of $\dfrac{7x+1}{(x+1) ^{4}}$ into partial fractions has three terms: $\dfrac{A}{x+1}+\dfrac{B}{(x+1) ^{2}}+\dfrac{C}{(x+1) ^{3}},$ where $A,$ $B,$ and $C$ are real numbers.

False

Skill Building

In Problems 5–8, find each integral by first writing the integrand as the sum of a polynomial and a proper rational function.

$\int \dfrac{x^{2}+1}{x+1}dx$

$\dfrac{x^2}{2}-x+2\ln\left|x+1\right|+C$

$\int \dfrac{x^{2}+4}{x-2}dx$

$\int \dfrac{x^{3}+3x-4}{x-2}dx$

$\dfrac{x^3}{3}+x^2+7x+10\ln\left|x-2\right|+C$

$\int \dfrac{x^{3}-3x^{2}+4}{x+3}dx$

In Problems 9–14, find each integral. (Hint: Each of the denominators contains only distinct linear factors.)

$\int \dfrac{dx}{(x-2)(x+1)}$

$\dfrac{1}{3}\ln\left|x-2\right| - \dfrac{1}{3}\ln\left|x+1\right| + C$

$\int \dfrac{dx}{(x+4)(x-1)}$

$\int \dfrac{x\,dx}{(x-1)(x-2)}$

$-\ln\left|x-1\right| + 2\ln\left|x-2\right| + C$

$\int \dfrac{3x\,dx}{(x+2)(x-4)}$

$\int \dfrac{x\,dx}{(3x-2)(2x+1)}$

$\dfrac{2}{21}\ln\left|3x-2\right|+\dfrac{1}{14}\ln\left|2x+1\right|+C$

$\int \dfrac{dx}{(2x+3)(4x-1)}$

In Problems 15–18, find each integral. (Hint: Each of the denominators contains a repeated linear factor.)

$\int \dfrac{x-3}{(x+2)(x+1)^{2}}\,dx$

$-5\ln\left|x+2\right| + 5\ln\left|x+1\right| + \dfrac{4}{x+1}+C$

$\int \dfrac{x+1}{x^{2}(x-2)}\,dx$

$\int \dfrac{x^{2}\,dx}{(x-1)^{2}(x+1)}$

$\dfrac{1}{4}\ln\left|x+1\right| + \dfrac{3}{4}\ln\left|x-1\right| - \dfrac{1}{2(x-1)} + C$

$\int \dfrac{x^{2}+x}{(x+2)(x-1)^{2}}\,dx$

In Problems 19–22, find each integral. (Hint: Each of the denominators contains an irreducible quadratic factor.)

$\int \dfrac{dx}{x(x^{2}+1)}$

$\ln\left|x\right| -\dfrac{1}{2}\ln(x^{2}+1)+C$

$\int \dfrac{dx}{(x+1)(x^{2}+4)}$

$\int \dfrac{x^{2}+2x+3}{(x+1)(x^{2}+2x+4)}dx$

$\dfrac{1}{6}\ln(x^2+2x+4)+\dfrac{2}{3}\ln\left|x+1\right|+C$

$\int \dfrac{x^{2}-11x-18}{x(x^{2}+3x+3)}dx$

In Problems 23–26, find each integral. (Hint: Each of the denominators contains a repeated irreducible quadratic factor.)

$\int \dfrac{2x+1}{(x^{2}+16)^{2}}dx$

$-\dfrac{1}{x^{2}+16} + \dfrac{1}{128}\tan^{-1}\dfrac{x}{4} + \dfrac{x}{32(x^{2}+16)} + C$

$\int \dfrac{x^{2}+2x+3}{(x^{2}+4)^{2}}dx$

$\int \dfrac{x^{3}\,dx}{(x^{2}+16)^{3}}$

$-\dfrac{1}{2(x^{2}+16)} + \dfrac{4}{(x^{2}+ 16)^{2}} + C$

$\int \dfrac{x^{2}\,dx}{(x^{2}+4)^{3}}$

In Problems 27–36, find each integral.

$\int \dfrac{x\,dx}{x^{2}+2x-3}$

$\dfrac{1}{4}\ln\left|x-1\right| + \dfrac{3}{4}\ln\left|x+3\right| + C$

$\int \dfrac{x^{2}-x-8}{(x+1)(x^{2}+5x+6)}\,dx$

$\int \dfrac{10x^{2}+2x}{(x-1)^{2}(x^{2}+2)}dx$

$\dfrac{14}{3}\ln\left|x-1\right| - \dfrac{4}{x-1} - \dfrac{7}{3}\ln(x^{2}+2) + \dfrac{2\sqrt{2}}{3}\tan^{-1}\dfrac{\sqrt{2}x}{2} + C$

$\int \dfrac{x+4}{x^{2}(x^{2}+4)}\,dx$

$\int \dfrac{7x+3}{x^{3}-2x^{2}-3x}\,dx$

$\int \dfrac{x^{5}+1}{x^{6}-x^{4}}\,dx$

$\int \dfrac{x^{2}}{(x-2) (x-1) ^{2}}dx$

$4\ln\left|x-2\right| - 3\ln\left|x-1\right| + \dfrac{1}{x-1} + C$

$\int \dfrac{x^{2}+1}{(x+3) (x-1) ^{2}}\,dx$

$\int \dfrac{2x+1}{x^{3}-1}dx$

$\ln\left|x-1\right| - \dfrac{1}{2}\ln(x^{2}+x+1) + \dfrac{\sqrt{3}}{3}\tan^{-1}\dfrac{\sqrt{3}(2x+1)}{3} + C$

$\int \dfrac{dx}{x^{3}-8}$

In Problems 37–40, find each definite integral.

$\int_{0}^{1}\dfrac{dx}{x^{2}-9}$

$\dfrac{1}{6}\ln\dfrac{1}{2}$

$\int_{2}^{4}\dfrac{dx}{x^{2}-25}$

$\int_{-2}^{3}\dfrac{dx}{16-x^{2}}$

$\dfrac{1}{8}\ln21$

$\int_{1}^{2}\dfrac{dx}{9-x^{2}}$

Applications and Extensions

In Problems 41–54, find each integral. (Hint: Make a substitution before using partial fraction decomposition.)

$\int \dfrac{\cos \theta \,}{\sin ^{2}\theta +\sin \theta -6}\,d\theta$

$\dfrac{1}{5}\ln(2-\sin\theta) - \dfrac{1}{5}\ln(\sin\theta + 3) +C$

$\int \dfrac{\sin x}{\cos ^{2}x-2\cos x-8}dx$

$\int \dfrac{\sin \theta \,}{\cos ^{3}\theta +\cos \theta }\,d\theta$

$-\ln\left|\cos\theta\right| + \dfrac{1}{2}\ln(\cos^{2}\theta + 1) + C$

$\int \dfrac{4\cos \theta \,}{\sin ^{3}\theta +2\sin \theta }\,d\theta$

$\int \dfrac{e^{t}\,}{e^{2t}+e^{t}-2}\,dt$

$\dfrac{1}{3}\ln\left|e^{t}-1\right| - \dfrac{1}{3}\ln(e^{t}+2) + C$

$\int \dfrac{e^{x}}{e^{2x}+e^{x}-6}\,dx$

$\int \dfrac{e^{x}}{e^{2x}-1}dx$

$\dfrac{1}{2}\ln\left|e^{x}-1\right| - \dfrac{1}{2}\ln(e^{x}+1) + C$

$\int \dfrac{dx}{e^{x}-e^{-x}}$

$\int \dfrac{dt}{e^{2t}+1}$

$t-\dfrac{1}{2}\ln(e^{2t}+1)+C$

$\int \dfrac{dt}{e^{3t}+e^{t}}$

$\int \dfrac{\sin x\cos x}{(\sin x - 1)^2}\,dx$

$\ln\left|\sin x - 1\right| - \dfrac{1}{\sin x - 1}+C$

$\int \dfrac{\cos x\sin x\,}{(\cos x-2)^2}\,dx$

$\int\dfrac{\cos x}{ (\sin^2 x+9)^2 }\,dx$

$\dfrac{1}{54}\tan^{-1}\dfrac{\sin x}{3} + \dfrac{1}{18}\dfrac{\sin x}{\sin^{2}x + 9} + C$

$\int \dfrac{\sin x}{ (\cos^2 x+4)^2 } \,dx$

507

Area Find the area under the graph of $y=\dfrac{4}{x^{2}-4}$ from $x=3$ to $x=5$ , as shown in the figure below.

$\ln\dfrac{15}{7}$

Area Find the area under the graph of $y=\dfrac{x-4}{(x+3) ^{2}}$ from $x=4$ to $x=6$ , as shown in the figure below.

Area Find the area under the graph of $y=\dfrac{8}{x^{3}+1}$ from $x=0$ to $x=2$ , as shown in the figure below.

$\dfrac{4\pi\sqrt{3}}{3}+\dfrac{4}{3}\ln3$

Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graph of $y=\dfrac{x}{x^{2}-4}$ and the $x$ -axis from $x=3$ to $x=5$ about the $x$ -axis, as shown in the figure below.

1. Find the zeros of $q( x) =x^{3}+3x^{2}-10x-24.$
2. Factor $q.$
3. Find the integral $\int \dfrac{3x-7}{x^{3}+3x^{2}-10x-24}dx.$

$-4, -2, 3$
$(x+4)(x+2)(x-3)$
$\dfrac{13}{10}\ln\left|x+2\right| + \dfrac{2}{35}\ln\left|x-3\right| - \dfrac{19}{14}\ln\left|x+4\right|+C$

Challenge Problems

In Problems 60–71, simplify each integrand as follows: If the integrand involves fractional powers such as $x^{p/q}$ and $x^{r/s}$ , make the substitution $x=u^{n}$ , where $n$ is the least common denominator of $\dfrac{p}{q}$ and $\dfrac{r}{s}$ . Then find each integral.

$\int \dfrac{x\,dx}{3+\sqrt{x}}$

$\int \dfrac{dx}{\sqrt{x}+2}$

$2\sqrt{x}-4\,\ln\big(\sqrt{x}+2\big)+C$

$\int \dfrac{dx}{x-\sqrt[3]{x}}$

$\int \dfrac{x\,dx}{\sqrt[3]{x}-1}$

$\dfrac{3}{5}x^{5/3} + \dfrac{3}{4}x^{4/3}+\dfrac{3}{2}x^{2/3}+x+3x^{1/3} + 3\ln\left|x^{1/3}-1\right|+C$

$\int \dfrac{dx}{\sqrt{x}+\sqrt[3]{x}}$

$\int \dfrac{dx}{3\sqrt{x}-\sqrt[3]{x}}$

$\dfrac{2}{3}x^{1/2} + \dfrac{1}{3}x^{1/3} + \dfrac{2}{9}x^{1/6} + \dfrac{2}{27} \ln\left|3x^{1/6}-1\right|+C$

$\int \dfrac{dx}{\sqrt[3]{2+3x}}$

$\int \dfrac{dx}{\sqrt[4]{1+2x}}$

$\dfrac{2}{3}(2x+1)^{3/4}+C$

$\int \dfrac{x\,dx}{(1+x)^{3/4}}$

$\int \dfrac{dx}{(1+x)^{2/3}}$

$3(1+x)^{1/3}+C$

$\int \dfrac{\sqrt[3]{x}+1}{\sqrt[3]{x}-1}dx$

$\int \dfrac{dx}{\sqrt{x}(1+\sqrt[3]{x})^{2}}$

$-\dfrac{3x^{1/6}}{x^{1/3}+1}+3\tan^{-1}x^{1/6}+C$

Weierstrass Substitution In Problems 72–87, use the following substitution, called a Weierstrass substitution. If an integrand is a rational expression of $\sin x$ or $\cos x$ or both, the substitution $\begin{equation*} z=\tan \dfrac{x}{2}\qquad -\dfrac{\pi }{2} < \dfrac{x}{2} < \dfrac{\pi }{2} \end{equation*}$

or equivalently, $\begin{equation*} \sin x=\dfrac{2z}{1+z^{2}}\qquad \cos x=\dfrac{1-z^{2}}{1+z^{2}}\qquad dx=\dfrac{2\,dz}{1+z^{2}} \end{equation*}$

will transform the integrand into a rational function of $z$ .

$\int \dfrac{dx}{1-\sin x}$

$\int \dfrac{dx}{1+\sin x}$

$-\dfrac{2}{1+\tan \dfrac{x}{2}}+C$

$\int \dfrac{dx}{1-\cos x}$

$\int \dfrac{dx}{3+2\cos x}$

$\dfrac{2\sqrt{5}}{5}\tan^{-1}\left(\dfrac{\sqrt{5}}{5}\tan\dfrac{x}{2}\right)+C$

$\int \dfrac{2\,dx}{\sin x+\cos x}$

$\int \dfrac{dx}{1-\sin x+\cos x}$

$-\ln\left|1-\tan\dfrac{x}{2}\right|+C$

$\int \dfrac{\sin x}{3+\cos x}dx$

$\int \dfrac{dx}{\tan x-1}$

$\dfrac{1}{2}\ln\left|\tan^{2}\dfrac{x}{2}+2\tan\dfrac{x}{2}-1\right| - \dfrac{1}{2}\ln\left(\tan^{2}\dfrac{x}{2}+1\right) - \dfrac{x}{2} + C$

$\int \dfrac{dx}{\tan x-\sin x}$

$\int \dfrac{\sec x}{\tan x-2}dx$

$\dfrac{\sqrt{5}}{5}\ln\left|\tan\dfrac{x}{2}-\dfrac{\sqrt{5}}{2}+\dfrac{1}{2}\right| - \dfrac{\sqrt{5}}{2}\ln\left|\tan\dfrac{x}{2}+\dfrac{\sqrt{5}}{2}+\dfrac{1}{2}\right| + C$

$\int \dfrac{\cot x\,}{1+\sin x}dx$

$\int \dfrac{\sec x\,}{1+\sin x}dx$

$-\dfrac{1}{2}\ln\left|\tan\dfrac{x}{2}-1\right| + \dfrac{1}{2}\ln\left|\tan\dfrac{x}{2}+1\right| + \dfrac{1}{\tan\dfrac{x}{2}+1} - \dfrac{1}{\left(\tan\dfrac{x}{2}+1\right)^{2}} + C$

$\int_{0}^{\pi /2}\dfrac{dx}{\sin x+1}$

$\int_{\pi /4}^{\pi /3}\dfrac{\csc x\,}{3+4\tan x}\,dx$

$\dfrac{1}{3}\ln\dfrac{\sqrt{3}}{3} - \dfrac{1}{3}\ln(\sqrt{2}-1) - \dfrac{4}{15}\ln(\sqrt{3}+1) - \dfrac{4}{15}\ln(4-\sqrt{2}) + \dfrac{4}{15}\ln(3\sqrt{2}-2) + \dfrac{4}{15}\ln\left(3-\dfrac{\sqrt{3}}{3}\right)$

$\int_{0}^{\pi /2}\dfrac{\cos x\,}{2-\cos x}\,dx$

$\int_{0}^{\pi /4}\dfrac{4\,dx}{\tan x+1}$

$\dfrac{\pi}{2}+\ln2$

508

Use a Weierstrass substitution to derive the formula $\int \csc x\,dx=\ln \sqrt{\dfrac{1-\cos x}{1+\cos x}}+C$

Show that the result obtained in Problem 88 is equivalent to $\int \csc x\,dx=\ln \,\vert \csc x-\cot x\vert +C$

See the Student Solutions Manual.

Since $\dfrac{d}{dx}\tanh ^{-1}x=\dfrac{1}{1-x^{2}}$ , we might expect that $\int_{2}^{3}\dfrac{dx}{1-x^{2}} = \tanh ^{-1}3-\tanh ^{-1}2.$ Why is this incorrect? What is the correct result?

Show that the two formulas below are equivalent. $\begin{eqnarray*} \int \sec x\,dx &=& \ln \,\vert \sec x+\tan x \vert +C\\ \int \sec x\,dx &=& \ln \left\vert \dfrac{1+\tan \dfrac{x}{2} }{1-\tan \dfrac{x}{2}}\right\vert +C \end{eqnarray*} \left[ {Hint:} \tan \dfrac{x}{2}=\dfrac{\sin \dfrac{x }{2}}{\cos \dfrac{x}{2}}=\dfrac{\sin ^{2}\left( \dfrac{x}{2}\right) }{\sin \left( \dfrac{x}{2}\right) \cos \left( \dfrac{x}{2}\right) }=\dfrac{1-\cos x }{\sin x}.\right]$

See the Student Solutions Manual.

Use the methods of this section to find $\int \dfrac{dx}{1+x^{4} }$ . (Hint: Factor $1+x^{4}$ into irreducible quadratics.)

7.5 Assess Your UnderstandingPrinted Page 506

7.5 Assess Your Understanding
Printed Page 506