7.1E: Exercises
- Page ID
- 128838
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Use a table of integrals to evaluate the following integrals.
1) \(\displaystyle \int _0^4\frac{x}{\sqrt{1+2x}}\,dx\)
2) \(\displaystyle \int \frac{x+3}{x^2+2x+2}\,dx\)
- Answer
- \(\displaystyle \int \frac{x+3}{x^2+2x+2}\,dx = \tfrac{1}{2}\ln |x^2+2x+2|+2\arctan(x+1)+C\)
3) \(\displaystyle \int x^3\sqrt{1+2x^2}\,dx\)
4) \(\displaystyle \int \frac{1}{\sqrt{x^2+6x}}\,dx\)
- Answer
- \(\displaystyle \int \frac{1}{\sqrt{x^2+6x}}\,dx = \cosh^{−1}\left(\frac{x+3}{3}\right)+C\)
5) \(\displaystyle \int \frac{x}{x+1}\,dx\)
6) \(\displaystyle \int x \cdot 2^{x^2}\,dx\)
- Answer
- \(\displaystyle \int x \cdot 2^{x^2}\,dx = \frac{2^{x^2−1}}{\ln 2}+C\)
7) \(\displaystyle \int \frac{1}{4x^2+25}\,dx\)
8) \(\displaystyle \int \frac{dy}{\sqrt{4−y^2}}\)
- Answer
- \(\displaystyle \int \frac{dy}{\sqrt{4−y^2}} = \arcsin\left(\frac{y}{2}\right)+C\)
9) \(\displaystyle \int \sin^3(2x)\cos(2x)\,dx\)
10) \(\displaystyle \int \csc(2w)\cot(2w)\,dw\)
- Answer
- \(\displaystyle \int \csc(2w)\cot(2w)\,dw = −\tfrac{1}{2}\csc(2w)+C\)
11) \(\displaystyle \int 2^y\,dy\)
12) \(\displaystyle \int ^1_0\frac{3x}{\sqrt{x^2+8}}\,dx\)
- Answer
- \(\displaystyle \int ^1_0\frac{3x}{\sqrt{x^2+8}}\,dx = 9−6\sqrt{2}\)
13) \(\displaystyle \int ^{1/4}_{−1/4}\sec^2( \pi x)\tan( \pi x)\,dx\)
14) \(\displaystyle \int ^{ \pi /2}_0\tan^2\left(\frac{x}{2}\right)\,dx\)
- Answer
- \(\displaystyle \int ^{ \pi /2}_0\tan^2\left(\frac{x}{2}\right)\,dx = 2−\frac{ \pi }{2}\)
15) \(\displaystyle \int \cos^3x\,dx\)
16) \(\displaystyle \int \tan^5(3x)\,dx\)
- Answer
- \(\displaystyle \int \tan^5(3x)\,dx = \tfrac{1}{12}\tan^4(3x)−\tfrac{1}{6}\tan^2(3x)+\tfrac{1}{3}\ln|\sec 3x|+C\)
17) \(\displaystyle \int \sin^2y\cos^3y\,dy\)
Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers.
18) [Technology Required] \(\displaystyle \int \frac{dw}{1+\sec\left(\frac{w}{2}\right)}\)
- Answer
- \(\displaystyle \int \frac{dw}{1+\sec\left(\frac{w}{2}\right)} = 2\cot\left(\tfrac{w}{2}\right)−2\csc\left(\tfrac{w}{2}\right)+w+C\)
19) [Technology Required] \(\displaystyle \int \frac{dw}{1−\cos(7w)}\)
20) [Technology Required] \(\displaystyle \int ^t_0\frac{dt}{4\cos t+3\sin t}\)
- Answer
- \(\displaystyle \int ^t_0\frac{dt}{4\cos t+3\sin t} = \tfrac{1}{5}\ln\Big|\frac{2(5+4\sin t−3\cos t)}{4\cos t+3\sin t}\Big|\)
21) [Technology Required] \(\displaystyle \int \frac{\sqrt{x^2−9}}{3x}\,dx\)
22) [Technology Required] \(\displaystyle \int \frac{dx}{x^{1/2}+x^{1/3}}\)
- Answer
- \(\displaystyle \int \frac{dx}{x^{1/2}+x^{1/3}} = 6x^{1/6}−3x^{1/3}+2\sqrt{x}−6\ln[1+x^{1/6}]+C\)
23) [Technology Required] \(\displaystyle \int \frac{dx}{x\sqrt{x−1}}\)
24) [Technology Required] \(\displaystyle \int x^3\sin x\,dx\)
- Answer
- \(\displaystyle \int x^3\sin x\,dx = −x^3\cos x+3x^2\sin x+6x\cos x−6\sin x+C\)
25) [Technology Required] \(\displaystyle \int x\sqrt{x^4−9}\,dx\)
26) [Technology Required] \(\displaystyle \int \frac{x}{1+e^{−x^2}}\,dx\)
- Answer
- \(\displaystyle \int \frac{x}{1+e^{−x^2}}\,dx = \tfrac{1}{2}\left(x^2+\ln|1+e^{−x^2}|\right)+C\)
27) [Technology Required] \(\displaystyle \int \frac{\sqrt{3−5x}}{2x}\,dx\)
28) [Technology Required] \(\displaystyle \int \frac{dx}{x\sqrt{x−1}}\)
- Answer
- \(\displaystyle \int \frac{dx}{x\sqrt{x−1}} = 2\arctan\big(\sqrt{x−1}\big)+C\)
29) [Technology Required] \(\displaystyle \int e^x\cos^{−1}(e^x)\,dx\)
Use a calculator or CAS to evaluate the following integrals.
30) [Technology Required] \(\displaystyle \int ^{ \pi /4}_0\cos 2x \, dx\)
- Answer
- \(\displaystyle \int ^{ \pi /4}_0\cos 2x \, dx = 0.5=\frac{1}{2}\)
31) [Technology Required] \(\displaystyle \int ^1_0x \cdot e^{−x^2}\,dx\)
32) [Technology Required] \(\displaystyle \int ^8_0\frac{2x}{\sqrt{x^2+36}}\,dx\)
- Answer
- \(\displaystyle \int ^8_0\frac{2x}{\sqrt{x^2+36}}\,dx = 8.0\)
33) [Technology Required] \(\displaystyle \int ^{2/\sqrt{3}}_0\frac{1}{4+9x^2}\,dx\)
34) [Technology Required] \(\displaystyle \int \frac{dx}{x^2+4x+13}\)
- Answer
- \(\displaystyle \int \frac{dx}{x^2+4x+13} = \tfrac{1}{3}\arctan\left(\tfrac{1}{3}(x+2)\right)+C\)
35) [Technology Required] \(\displaystyle \int \frac{dx}{1+\sin x}\)
Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table.
36) \(\displaystyle \int \frac{dx}{x^2+2x+10}\)
- Answer
- \(\displaystyle \int \frac{dx}{x^2+2x+10} = \tfrac{1}{3}\arctan\left(\frac{x+1}{3}\right)+C\)
37) \(\displaystyle \int \frac{dx}{\sqrt{x^2−6x}}\)
38) \(\displaystyle \int \frac{e^x}{\sqrt{e^{2x}−4}}\,dx\)
- Answer
- \(\displaystyle \int \frac{e^x}{\sqrt{e^{2x}−4}}\,dx = \ln\left(e^x+\sqrt{4+e^{2x}}\right)+C\)
39) \(\displaystyle \int \frac{\cos x}{\sin^2x+2\sin x}\,dx\)
40) \(\displaystyle \int \frac{\arctan(x^3)}{x^4}\,dx\)
- Answer
- \(\displaystyle \int \frac{\arctan(x^3)}{x^4}\,dx = \ln x−\tfrac{1}{6}\ln(x^6+1)−\frac{\arctan(x^3)}{3x^3}+C\)
41) \(\displaystyle \int \frac{\ln|x|\arcsin\left(\ln|x|\right)}{x}\,dx\)
Use tables to perform the integration.
42) \(\displaystyle \int \frac{dx}{\sqrt{x^2+16}}\)
- Answer
- \(\displaystyle \int \frac{dx}{\sqrt{x^2+16}} = \ln |x|+\sqrt{16+x^2}∣+C\)
43) \(\displaystyle \int \frac{3x}{2x+7}\,dx\)
44) \(\displaystyle \int \frac{dx}{1−\cos 4x}\)
- Answer
- \(\displaystyle \int \frac{dx}{1−\cos 4x} = −\frac{1}{4}\cot 2x+C\)
45) \(\displaystyle \int \frac{dx}{\sqrt{4x+1}}\)
46) Find the area bounded by \(y(4+25x^2)=5,\;x=0,\;y=0,\) and \(x=4.\) Use a table of integrals or a CAS.
- Answer
- \(\frac{1}{2}\arctan 10\) units²
47) The region bounded between the curve \(y=\dfrac{1}{\sqrt{1+\cos x}}, \; 0.3 \leq x \leq 1.1,\) and the \(x\)-axis is revolved about the \(x\)-axis to generate a solid. Use a table of integrals to find the volume of the solid generated. (Round the answer to two decimal places.)
48) Use substitution and a table of integrals to find the area of the surface generated by revolving the curve \(y=e^x,\; 0 \leq x \leq 3,\) about the \(x\)-axis. (Round the answer to two decimal places.)
- Answer
- \(1276.14\) units²
49) [Technology Required] Use an integral table and a calculator to find the area of the surface generated by revolving the curve \(y=\dfrac{x^2}{2},\; 0 \leq x \leq 1,\) about the \(x\)-axis. (Round the answer to two decimal places.)
50) [Technology Required] Use a CAS or tables to find the area of the surface generated by revolving the curve \(y=\cos x,\; 0 \leq x \leq \frac{ \pi }{2},\) about the \(x\)-axis. (Round the answer to two decimal places.)
- Answer
- \(7.21\) units²
51) Find the length of the curve \(y=\dfrac{x^2}{4}\) over \([0,8]\).
52) Find the length of the curve \(y=e^x\) over \([0,\,\ln(2)].\)
- Answer
- \(\left(\sqrt{5}−\sqrt{2}+\ln\Big|\frac{2+2\sqrt{2}}{1+\sqrt{5}}\Big|\right)\) units
53) Find the area of the surface formed by revolving the graph of \(y=2\sqrt{x}\) over the interval \([0,9]\) about the \(x\)-axis.
54) Find the average value of the function \(f(x)=\dfrac{1}{x^2+1}\) over the interval \([−3,3].\)
- Answer
- \(\frac{1}{3}\arctan(3) \approx 0.416\)
55) Approximate the arc length of the curve \(y=\tan \pi x\) over the interval \(\left[0,\frac{1}{4}\right]\). (Round the answer to three decimal places.)
Contributors
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.