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Mathematics LibreTexts

7.5E: Exercises for Section 7.5

  • Page ID
    70433
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    Use a table of integrals to evaluate the following integrals.

    1) \(\displaystyle ∫_0^4\frac{x}{\sqrt{1+2x}}\,dx\)

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

    Answer
    \(\displaystyle ∫\frac{x+3}{x^2+2x+2}\,dx = \tfrac{1}{2}\ln |x^2+2x+2|+2\arctan(x+1)+C\)

    3) \(\displaystyle ∫x^3\sqrt{1+2x^2}\,dx\)

    4) \(\displaystyle ∫\frac{1}{\sqrt{x^2+6x}}\,dx\)

    Answer
    \(\displaystyle ∫\frac{1}{\sqrt{x^2+6x}}\,dx = \cosh^{−1}\left(\frac{x+3}{3}\right)+C\)

    5) \(\displaystyle ∫\frac{x}{x+1}\,dx\)

    6) \(\displaystyle ∫x⋅2^{x^2}\,dx\)

    Answer
    \(\displaystyle ∫x⋅2^{x^2}\,dx = \frac{2^{x^2−1}}{\ln 2}+C\)

    7) \(\displaystyle ∫\frac{1}{4x^2+25}\,dx\)

    8) \(\displaystyle ∫\frac{dy}{\sqrt{4−y^2}}\)

    Answer
    \(\displaystyle ∫\frac{dy}{\sqrt{4−y^2}} = \arcsin\left(\frac{y}{2}\right)+C\)

    9) \(\displaystyle ∫\sin^3(2x)\cos(2x)\,dx\)

    10) \(\displaystyle ∫\csc(2w)\cot(2w)\,dw\)

    Answer
    \(\displaystyle ∫\csc(2w)\cot(2w)\,dw = −\tfrac{1}{2}\csc(2w)+C\)

    11) \(\displaystyle ∫2^y\,dy\)

    12) \(\displaystyle ∫^1_0\frac{3x}{\sqrt{x^2+8}}\,dx\)

    Answer
    \(\displaystyle ∫^1_0\frac{3x}{\sqrt{x^2+8}}\,dx = 9−6\sqrt{2}\)

    13) \(\displaystyle ∫^{1/4}_{−1/4}\sec^2(πx)\tan(πx)\,dx\)

    14) \(\displaystyle ∫^{π/2}_0\tan^2\left(\frac{x}{2}\right)\,dx\)

    Answer
    \(\displaystyle ∫^{π/2}_0\tan^2\left(\frac{x}{2}\right)\,dx = 2−\frac{π}{2}\)

    15) \(\displaystyle ∫\cos^3x\,dx\)

    16) \(\displaystyle ∫\tan^5(3x)\,dx\)

    Answer
    \(\displaystyle ∫\tan^5(3x)\,dx = \tfrac{1}{12}\tan^4(3x)−\tfrac{1}{6}\tan^2(3x)+\tfrac{1}{3}\ln|\sec 3x|+C\)

    17) \(\displaystyle ∫\sin^2y\cos^3y\,dy\)

     

    Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers.

    18) [T] \(\displaystyle ∫\frac{dw}{1+\sec\left(\frac{w}{2}\right)}\)

    Answer
    \(\displaystyle ∫\frac{dw}{1+\sec\left(\frac{w}{2}\right)} = 2\cot\left(\tfrac{w}{2}\right)−2\csc\left(\tfrac{w}{2}\right)+w+C\)

    19) [T] \(\displaystyle ∫\frac{dw}{1−\cos(7w)}\)

    20) [T] \(\displaystyle ∫^t_0\frac{dt}{4\cos t+3\sin t}\)

    Answer
    \(\displaystyle ∫^t_0\frac{dt}{4\cos t+3\sin t} = \tfrac{1}{5}\ln\Big|\frac{2(5+4\sin t−3\cos t)}{4\cos t+3\sin t}\Big|\)

    21) [T] \(\displaystyle ∫\frac{\sqrt{x^2−9}}{3x}\,dx\)

    22) [T] \(\displaystyle ∫\frac{dx}{x^{1/2}+x^{1/3}}\)

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

    23) [T] \(\displaystyle ∫\frac{dx}{x\sqrt{x−1}}\)

    24) [T] \(\displaystyle ∫x^3\sin x\,dx\)

    Answer
    \(\displaystyle ∫x^3\sin x\,dx = −x^3\cos x+3x^2\sin x+6x\cos x−6\sin x+C\)

    25) [T] \(\displaystyle ∫x\sqrt{x^4−9}\,dx\)

    26) [T] \(\displaystyle ∫\frac{x}{1+e^{−x^2}}\,dx\)

    Answer
    \(\displaystyle ∫\frac{x}{1+e^{−x^2}}\,dx = \tfrac{1}{2}\left(x^2+\ln|1+e^{−x^2}|\right)+C\)

    27) [T] \(\displaystyle ∫\frac{\sqrt{3−5x}}{2x}\,dx\)

    28) [T] \(\displaystyle ∫\frac{dx}{x\sqrt{x−1}}\)

    Answer
    \(\displaystyle ∫\frac{dx}{x\sqrt{x−1}} = 2\arctan\big(\sqrt{x−1}\big)+C\)

    29) [T] \(\displaystyle ∫e^x\cos^{−1}(e^x)\,dx\)

     

    Use a calculator or CAS to evaluate the following integrals.

    30) [T] \(\displaystyle ∫^{π/4}_0\cos 2x \, dx\)

    Answer
    \(\displaystyle ∫^{π/4}_0\cos 2x \, dx = 0.5=\frac{1}{2}\)

    31) [T] \(\displaystyle ∫^1_0x⋅e^{−x^2}\,dx\)

    32) [T] \(\displaystyle ∫^8_0\frac{2x}{\sqrt{x^2+36}}\,dx\)

    Answer
    \(\displaystyle ∫^8_0\frac{2x}{\sqrt{x^2+36}}\,dx = 8.0\)

    33) [T] \(\displaystyle ∫^{2/\sqrt{3}}_0\frac{1}{4+9x^2}\,dx\)

    34) [T] \(\displaystyle ∫\frac{dx}{x^2+4x+13}\)

    Answer
    \(\displaystyle ∫\frac{dx}{x^2+4x+13} = \tfrac{1}{3}\arctan\left(\tfrac{1}{3}(x+2)\right)+C\)

    35) [T] \(\displaystyle ∫\frac{dx}{1+\sin x}\)

     

    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) \(\displaystyle ∫\frac{dx}{x^2+2x+10}\)

    Answer
    \(\displaystyle ∫\frac{dx}{x^2+2x+10} = \tfrac{1}{3}\arctan\left(\frac{x+1}{3}\right)+C\)

    37) \(\displaystyle ∫\frac{dx}{\sqrt{x^2−6x}}\)

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

    Answer
    \(\displaystyle ∫\frac{e^x}{\sqrt{e^{2x}−4}}\,dx = \ln\left(e^x+\sqrt{4+e^{2x}}\right)+C\)

    39) \(\displaystyle ∫\frac{\cos x}{\sin^2x+2\sin x}\,dx\)

    40) \(\displaystyle ∫\frac{\arctan(x^3)}{x^4}\,dx\)

    Answer
    \(\displaystyle ∫\frac{\arctan(x^3)}{x^4}\,dx = \ln x−\tfrac{1}{6}\ln(x^6+1)−\frac{\arctan(x^3)}{3x^3}+C\)

    41) \(\displaystyle ∫\frac{\ln|x|\arcsin\left(\ln|x|\right)}{x}\,dx\)

     

    Use tables to perform the integration.

    42) \(\displaystyle ∫\frac{dx}{\sqrt{x^2+16}}\)

    Answer
    \(\displaystyle ∫\frac{dx}{\sqrt{x^2+16}} = \ln |x|+\sqrt{16+x^2}∣+C\)

    43) \(\displaystyle ∫\frac{3x}{2x+7}\,dx\)

    44) \(\displaystyle ∫\frac{dx}{1−\cos 4x}\)

    Answer
    \(\displaystyle ∫\frac{dx}{1−\cos 4x} = −\frac{1}{4}\cot 2x+C\)

    45) \(\displaystyle ∫\frac{dx}{\sqrt{4x+1}}\)

    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 \(y=\dfrac{1}{\sqrt{1+\cos x}}, \; 0.3≤x≤1.1,\) 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 \(y=e^x,\; 0≤x≤3,\) about the \(x\)-axis. (Round the answer to two decimal places.)

    Answer
    \(1276.14\) units²

    49) [T] Use an integral table and a calculator to find the area of the surface generated by revolving the curve \(y=\dfrac{x^2}{2},\; 0≤x≤1,\) about the \(x\)-axis. (Round the answer to two decimal places.)

    50) [T] Use a CAS or tables to find the area of the surface generated by revolving the curve \(y=\cos x,\; 0≤x≤\frac{π}{2},\) 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
    \(\frac{1}{3}\arctan(3)≈0.416\)

    55) Approximate the arc length of the curve \(y=\tan πx\) 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.

     


    7.5E: Exercises for Section 7.5 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.