
2.4E: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Partial Fractions

Use partial fraction decomposition (or a simpler technique) to express the rational function as a sum or difference of two or more simpler rational expressions.

Exercise $$\PageIndex{1}$$

$$\dfrac{1}{(x−3)(x−2)}$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{2}$$

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

$$\dfrac{x^2+1}{x(x+1)(x+2)} \quad = \quad −\dfrac{2}{x+1}+\dfrac{5}{2(x+2)}+\dfrac{1}{2x}$$

Exercise $$\PageIndex{3}$$

$$\dfrac{1}{x^3−x}$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{4}$$

$$\dfrac{3x+1}{x^2}$$

$$\dfrac{3x+1}{x^2} \quad = \quad \dfrac{1}{x^2}+\dfrac{3}{x}$$

Exercise $$\PageIndex{5}$$

$$\dfrac{3x^2}{x^2+1}$$ (Hint: Use long division first.)

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{6}$$

$$\dfrac{2x^4}{x^2−2x}$$

$$\dfrac{2x^4}{x^2−2x} \quad = \quad 2x^2+4x+8+\dfrac{16}{x−2}$$

Exercise $$\PageIndex{7}$$

$$\dfrac{1}{(x−1)(x^2+1)}$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{8}$$

$$\dfrac{1}{x^2(x−1)}$$

$$\dfrac{1}{x^2(x−1)} \quad = \quad −\dfrac{1}{x^2}−\dfrac{1}{x}+\dfrac{1}{x−1}$$

Exercise $$\PageIndex{9}$$

$$\dfrac{x}{x^2−4}$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{10}$$

$$\dfrac{1}{x(x−1)(x−2)(x−3)}$$

$$\dfrac{1}{x(x−1)(x−2)(x−3)} \quad = \quad −\dfrac{1}{2(x−2)}+\dfrac{1}{2(x−1)}−\dfrac{1}{6x}+\dfrac{1}{6(x−3)}$$

Exercise $$\PageIndex{11}$$

$$\dfrac{1}{x^4−1}=\dfrac{1}{(x+1)(x−1)(x^2+1)}$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{12}$$

$$\dfrac{3x^2}{x^3−1}=\dfrac{3x^2}{(x−1)(x^2+x+1)}$$

$$\dfrac{3x^2}{x^3−1} \quad = \quad \dfrac{1}{x−1}+\dfrac{2x+1}{x^2+x+1}$$

Exercise $$\PageIndex{13}$$

$$\dfrac{2x}{(x+2)^2}$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{14}$$

$$\dfrac{3x^4+x^3+20x^2+3x+31}{(x+1)(x^2+4)^2}$$

$$\dfrac{3x^4+x^3+20x^2+3x+31}{(x+1)(x^2+4)^2} \quad = \quad \dfrac{2}{x+1}+\dfrac{x}{x^2+4}−\dfrac{1}{(x^2+4)^2}$$

In exercises 15 - 25, use the method of partial fractions to evaluate each of the following integrals.

Exercise $$\PageIndex{15}$$

$$\displaystyle ∫\frac{dx}{(x−3)(x−2)}$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{16}$$

$$\displaystyle ∫\frac{3x}{x^2+2x−8}\,dx$$

$$\displaystyle ∫\frac{3x}{x^2+2x−8}\,dx \quad = \quad −\ln|2−x|+2\ln|4+x|+C = \ln\left| \frac{(4+x)^2}{2-x} \right| + C$$

Exercise $$\PageIndex{17}$$

$$\displaystyle ∫\frac{dx}{x^3−x}$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{18}$$

$$\displaystyle ∫\frac{x}{x^2−4}\,dx$$

$$\displaystyle ∫\frac{x}{x^2−4}\,dx \quad = \quad \tfrac{1}{2}\ln|4−x^2|+C$$

Exercise $$\PageIndex{19}$$

$$\displaystyle ∫\frac{dx}{x(x−1)(x−2)(x−3)}$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{20}$$

$$\displaystyle ∫\frac{2x^2+4x+22}{x^2+2x+10}\,dx$$

$$\displaystyle ∫\frac{2x^2+4x+22}{x^2+2x+10}\,dx \quad = \quad 2\left(x+\tfrac{1}{3}\arctan\left(\frac{1+x}{3}\right)\right)+C$$

Exercise $$\PageIndex{21}$$

$$\displaystyle ∫\frac{dx}{x^2−5x+6}$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{22}$$

$$\displaystyle ∫\frac{2−x}{x^2+x}\,dx$$

$$\displaystyle ∫\frac{2−x}{x^2+x}\,dx \quad = \quad 2\ln|x|−3\ln|1+x|+C = \ln\left| \frac{x^2}{(1+x)^3} \right|+C$$

Exercise $$\PageIndex{23}$$

$$\displaystyle ∫\frac{2}{x^2−x−6}\,dx$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{24}$$

$$\displaystyle ∫\frac{dx}{x^3−2x^2−4x+8}$$

$$\displaystyle ∫\frac{dx}{x^3−2x^2−4x+8} \quad = \quad \tfrac{1}{16}\left(−\frac{4}{−2+x}−\ln|−2+x|+\ln|2+x|\right)+C = \tfrac{1}{16}\left(−\frac{4}{−2+x}+\ln\left| \frac{x+2}{x-2} \right|\right)+C$$

Exercise $$\PageIndex{25}$$

$$\displaystyle ∫\frac{dx}{x^4−10x^2+9}$$

Add texts here. Do not delete this text first.

In exercises 26 - 29, evaluate the integrals with irreducible quadratic factors in the denominators.

Exercise $$\PageIndex{26}$$

$$\displaystyle ∫\frac{2}{(x−4)(x^2+2x+6)}\,dx$$

$$\displaystyle ∫\frac{2}{(x−4)(x^2+2x+6)}\,dx \quad = \quad \tfrac{1}{30}(−2\sqrt{5}\arctan\left[\frac{1+x}{\sqrt{5}}\right]+2\ln|−4+x|−\ln|6+2x+x^2|)+C$$

Exercise $$\PageIndex{27}$$

$$\displaystyle ∫\frac{x^2}{x^3−x^2+4x−4}\,dx$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{28}$$

$$\displaystyle ∫\frac{x^3+6x^2+3x+6}{x^3+2x^2}\,dx$$

$$\displaystyle ∫\frac{x^3+6x^2+3x+6}{x^3+2x^2}\,dx \quad = \quad −\frac{3}{x}+4\ln|x+2|+x+C$$

Exercise $$\PageIndex{29}$$

$$\displaystyle ∫\frac{x}{(x−1)(x^2+2x+2)^2}\,dx$$

Add texts here. Do not delete this text first.

In exercises 30 - 32, use the method of partial fractions to evaluate the integrals.

Exercise $$\PageIndex{30}$$

$$\displaystyle ∫\frac{3x+4}{(x^2+4)(3−x)}\,dx$$

$$\displaystyle ∫\frac{3x+4}{(x^2+4)(3−x)}\,dx \quad = \quad −\ln|3−x|+\tfrac{1}{2}\ln|x^2+4|+C$$

Exercise $$\PageIndex{31}$$

$$\displaystyle ∫\frac{2}{(x+2)^2(2−x)}\,dx$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{32}$$

$$\displaystyle ∫\frac{3x+4}{x^3−2x−4}\,dx$$ (Hint: Use the rational root theorem.)

$$\displaystyle ∫\frac{3x+4}{x^3−2x−4}\,dx \quad = \quad \ln|x−2|−\tfrac{1}{2}\ln|x^2+2x+2|+C$$

In exercises 33 - 46, use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals.

Exercise $$\PageIndex{33}$$

$$\displaystyle ∫^1_0\frac{e^x}{36−e^{2x}}\,dx$$ (Give the exact answer and the decimal equivalent. Round to five decimal places.)

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{34}$$

$$\displaystyle ∫\frac{e^x\,dx}{e^{2x}−e^x}\,dx$$

$$\displaystyle ∫\frac{e^x\,dx}{e^{2x}−e^x}\,dx \quad = \quad −x+\ln|1−e^x|+C$$

Exercise $$\PageIndex{35}$$

$$\displaystyle ∫\frac{\sin x\,dx}{1−\cos^2x}$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{36}$$

$$\displaystyle ∫\frac{\sin x}{\cos^2 x+\cos x−6}\,dx$$

$$\displaystyle ∫\frac{\sin x}{\cos^2 x+\cos x−6}\,dx \quad = \quad \tfrac{1}{5}\ln\left|\frac{\cos x+3}{\cos x−2}\right|+C$$

Exercise $$\PageIndex{37}$$

$$\displaystyle ∫\frac{1−\sqrt{x}}{1+\sqrt{x}}\,dx$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{38}$$

$$\displaystyle ∫\frac{dt}{(e^t−e^{−t})^2}$$

$$\displaystyle ∫\frac{dt}{(e^t−e^{−t})^2} \quad = \quad \frac{1}{2−2e^{2t}}+C$$

Exercise $$\PageIndex{39}$$

$$\displaystyle ∫\frac{1+e^x}{1−e^x}\,dx$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{40}$$

$$\displaystyle ∫\frac{dx}{1+\sqrt{x+1}}$$

$$\displaystyle ∫\frac{dx}{1+\sqrt{x+1}} \quad = \quad 2\sqrt{1+x}−2\ln|1+\sqrt{1+x}|+C$$

Exercise $$\PageIndex{41}$$

$$\displaystyle ∫\frac{dx}{\sqrt{x}+\sqrt[4]{x}}$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{42}$$

$$\displaystyle ∫\frac{\cos x}{\sin x(1−\sin x)}\,dx$$

$$\displaystyle ∫\frac{\cos x}{\sin x(1−\sin x)}\,dx \quad = \quad \ln\left|\frac{\sin x}{1−\sin x}\right|+C$$

Exercise $$\PageIndex{43}$$

$$\displaystyle ∫\frac{e^x}{(e^{2x}−4)^2}\,dx$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{44}$$

$$\displaystyle ∫_1^2\frac{1}{x^2\sqrt{4−x^2}}\,dx$$

$$\displaystyle ∫_1^2\frac{1}{x^2\sqrt{4−x^2}}\,dx \quad = \quad \frac{\sqrt{3}}{4}$$

Exercise $$\PageIndex{45}$$

$$\displaystyle ∫\frac{1}{2+e^{−x}}\,dx$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{46}$$

$$\displaystyle ∫\frac{1}{1+e^x}\,dx$$

$$\displaystyle ∫\frac{1}{1+e^x}\,dx \quad = \quad x−\ln(1+e^x)+C$$

In exercises 47 - 48, use the given substitution to convert the integral to an integral of a rational function, then evaluate.

Exercise $$\PageIndex{47}$$

$$\displaystyle ∫\frac{1}{t−\sqrt[3]{t}}\,dt; \quad t=x^3$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{48}$$

$$\displaystyle ∫\frac{1}{\sqrt{x}+\sqrt[3]{x}}\,dx; \quad x=u^6$$

$$\displaystyle ∫\frac{1}{\sqrt{x}+\sqrt[3]{x}}\,dx \quad = \quad 6x^{1/6}−3x^{1/3}+2\sqrt{x}−6\ln(1+x^{1/6})+C$$

Exercise $$\PageIndex{49}$$

Graph the curve $$y=\dfrac{x}{1+x}$$ over the interval $$[0,5]$$. Then, find the area of the region bounded by the curve, the $$x$$-axis, and the line $$x=4$$.

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{50}$$

Find the volume of the solid generated when the region bounded by $$y=\dfrac{1}{\sqrt{x(3−x)}}, \,y=0, \,x=1,$$ and $$x=2$$ is revolved about the $$x$$-axis.

$$V = \frac{4}{3}π\text{arctanh}\,\left[\frac{1}{3}\right]=\frac{1}{3}π\ln 4 \, \text{units}^3$$

Exercise $$\PageIndex{51}$$

The velocity of a particle moving along a line is a function of time given by $$v(t)=\dfrac{88t^2}{t^2+1}.$$ Find the distance that the particle has traveled after $$t=5$$ sec.

Add texts here. Do not delete this text first.

In exercises 52 - 54, solve the initial-value problem for $$x$$ as a function of $$t$$.

Exercise $$\PageIndex{52}$$

$$(t^2−7t+12)\dfrac{dx}{dt}=1,\quad t>4,\, x(5)=0$$

$$x=−\ln|t−3|+\ln|t−4|+\ln 2 = \ln\left| \dfrac{2(t-4)}{t-3}\right|$$

Exercise $$\PageIndex{53}$$

$$(t+5)\dfrac{dx}{dt}=x^2+1, \quad t>−5,\,x(1)=\tan 1$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{54}$$

$$(2t^3−2t^2+t−1)\dfrac{dx}{dt}=3,\quad x(2)=0$$

$$x=\ln|t−1|−\sqrt{2}\arctan(\sqrt{2}t)−\frac{1}{2}\ln(t^2+\frac{1}{2})+\sqrt{2}\arctan(2\sqrt{2})+\frac{1}{2}\ln 4.5$$

Exercise $$\PageIndex{55}$$

Find the $$x$$-coordinate of the centroid of the area bounded by $$y(x^2−9)=1, \, y=0, \,x=4,$$ and $$x=5.$$ (Round the answer to two decimal places.)

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{56}$$

Find the volume generated by revolving the area bounded by $$y=\dfrac{1}{x^3+7x^2+6x},\, x=1,\, x=7$$, and $$y=0$$ about the $$y$$-axis.

$$V = \frac{2}{5}π\ln\frac{28}{13} \, \text{units}^3$$

Exercise $$\PageIndex{57}$$

Find the area bounded by $$y=\dfrac{x−12}{x^2−8x−20}, \,y=0, \,x=2,$$ and $$x=4$$. (Round the answer to the nearest hundredth.)

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{58}$$

Evaluate the integral $$\displaystyle ∫\frac{dx}{x^3+1}.$$

$$\displaystyle ∫\frac{dx}{x^3+1} \quad = \quad \frac{\arctan[\frac{−1+2x}{\sqrt{3}}]}{\sqrt{3}}+\frac{1}{3}\ln|1+x|−\frac{1}{6}\ln∣1−x+x^2∣+C$$

For problems 59 - 62, use the substitutions $$\tan(\frac{x}{2})=t, \,dx=\dfrac{2}{1+t^2}\,dt, \, \sin x=\dfrac{2t}{1+t^2},$$ and $$\cos x=\dfrac{1−t^2}{1+t^2}.$$

Exercise $$\PageIndex{59}$$

$$\displaystyle ∫\frac{dx}{3−5\sin x}$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{60}$$

Find the area under the curve $$y=\dfrac{1}{1+\sin x}$$ between $$x=0$$ and $$x=π.$$ (Assume the dimensions are in inches.)

2.0 in.2

Exercise $$\PageIndex{61}$$

Given $$\tan(\frac{x}{2})=t,$$ derive the formulas $$dx=\dfrac{2}{1+t^2}dt, \,\sin x=\dfrac{2t}{1+t^2}$$, and $$\cos x=\dfrac{1−t^2}{1+t^2}.$$

Exercise $$\PageIndex{62}$$
Evaluate $$\displaystyle ∫\frac{\sqrt[3]{x−8}}{x}\,dx.$$
$$\displaystyle ∫\frac{\sqrt[3]{x−8}}{x}\,dx \quad = \quad 3(−8+x)^{1/3}−2\sqrt{3}\arctan\left[\frac{−1+(−8+x)^{1/3}}{\sqrt{3}}\right]−2\ln\left[2+(−8+x)^{1/3}\right]+\ln\left[4−2(−8+x)^{1/3}+(−8+x)^{2/3}\right]+C$$