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Mathematics LibreTexts

2.7E: Exercises

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    Other Strategies for Integration

    Use a table of integrals to evaluate the following integrals.

    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{2}\)

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

    Answer

    \(\displaystyle \frac{1}{2}ln∣x^2+2x+2∣+2arctan(x+1)+C\)

    Exercise \(\PageIndex{3}\)

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

    Answer

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    Exercise \(\PageIndex{4}\)

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

    Answer

    \(\displaystyle cosh^{−1}(\frac{x+3}{3})+C\)

    Exercise \(\PageIndex{5}\)

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

    Answer

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    Exercise \(\PageIndex{6}\)

    \(\displaystyle ∫x⋅2^{x^2}dx\)

    Answer

    \(\displaystyle \frac{2^{x^2−1}}{ln2}+C\)

    Exercise \(\PageIndex{7}\)

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

    Answer

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    Exercise \(\PageIndex{8}\)

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

    Answer

    \(\displaystyle arcsin(\frac{y}{2})+C\)

    Exercise \(\PageIndex{9}\)

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

    Answer

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    Exercise \(\PageIndex{10}\)

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

    Answer

    \(\displaystyle −\frac{1}{2}csc(2w)+C\)

    Exercise \(\PageIndex{11}\)

    \(\displaystyle ∫2^ydy\)

    Answer

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    Exercise \(\PageIndex{12}\)

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

    Answer

    \(\displaystyle 9−6\sqrt{2}\)

    Exercise \(\PageIndex{13}\)

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

    Answer

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    Exercise \(\PageIndex{14}\)

    \(\displaystyle ∫^{π/2}_0tan^2(\frac{x}{2})dx\)

    Answer

    \(\displaystyle 2−\frac{π}{2}\)

    Exercise \(\PageIndex{15}\)

    \(\displaystyle ∫cos^3xdx\)

    Answer

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    Exercise \(\PageIndex{16}\)

    \(\displaystyle ∫tan^5(3x)dx\)

    Answer

    \(\displaystyle \frac{1}{12}tan^4(3x)−\frac{1}{6}tan^2(3x)+\frac{1}{3}ln|sec(3x)|+C\)

    Exercise \(\PageIndex{17}\)

    \(\displaystyle ∫sin^2ycos^3ydy\)

    Answer

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

    Exercise \(\PageIndex{18}\)

    \(\displaystyle ∫\frac{dw}{1+sec(\frac{w}{2})}\)

    Answer

    \(\displaystyle 2cot(\frac{w}{2})−2csc(\frac{w}{2})+w+C\)

    Exercise \(\PageIndex{19}\)

    \(\displaystyle ∫\frac{dw}{1−cos(7w)}\)

    Answer

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    Exercise \(\PageIndex{20}\)

    \(\displaystyle ∫^t_0\frac{dt}{4cost+3sint}\)

    Answer

    \(\displaystyle \frac{1}{5}ln∣\frac{2(5+4sint−3cost)}{4cost+3sint}∣\)

    Exercise \(\PageIndex{21}\)

    \(\displaystyle ∫\frac{\sqrt{x^2−9}}{3x}dx\)

    Answer

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    Exercise \(\PageIndex{22}\)

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

    Answer

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

    Exercise \(\PageIndex{23}\)

    \(\displaystyle ∫\frac{dx}{x\sqrt{x−1}}\)

    Answer

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    Exercise \(\PageIndex{24}\)

    \(\displaystyle ∫x^3sinxdx\)

    Answer

    \(\displaystyle −x^3cosx+3x^2sinx+6xcosx−6sinx+C\)

    Exercise \(\PageIndex{25}\)

    \(\displaystyle ∫x\sqrt{x^4−9}dx\)

    Answer

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    Exercise \(\PageIndex{26}\)

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

    Answer

    \(\displaystyle \frac{1}{2}(x^2+ln∣1+e^{−x^2}∣)+C\)

    Exercise \(\PageIndex{27}\)

    \(\displaystyle ∫\frac{\sqrt{3−5x}}{2x}dx\)

    Answer

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    Exercise \(\PageIndex{28}\)

    \(\displaystyle ∫\frac{dx}{x\sqrt{x−1}}\)

    Answer

    \(\displaystyle 2arctan(\sqrt{x−1})+C\)

    Exercise \(\PageIndex{29}\)

    \(\displaystyle ∫e^xcos^{−1}(e^x)dx\)

    Answer

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    Use a calculator or CAS to evaluate the following integrals.

    Exercise \(\PageIndex{30}\)

    \(\displaystyle ∫^{π/4}_0cos(2x)dx\)

    Answer

    \(\displaystyle 0.5=\frac{1}{2}\)

    Exercise \(\PageIndex{31}\)

    \(\displaystyle ∫^1_0x⋅e^{−x^2}dx\)

    Answer

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    Exercise \(\PageIndex{32}\)

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

    Answer

    \(\displaystyle 8.0\)

    Exercise \(\PageIndex{33}\)

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

    Answer

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    Exercise \(\PageIndex{34}\)

    \(\displaystyle ∫\frac{dx}{x^2+4x+13}\)

    Answer

    \(\displaystyle \frac{1}{3}arctan(\frac{1}{3}(x+2))+C\)

    Exercise \(\PageIndex{35}\)

    \(\displaystyle ∫\frac{dx}{1+sinx}\)

    Answer

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    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.

    Exercise \(\PageIndex{36}\)

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

    Answer

    \(\displaystyle \frac{1}{3}arctan(\frac{x+1}{3})+C\)

    Exercise \(\PageIndex{37}\)

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

    Answer

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    Exercise \(\PageIndex{38}\)

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

    Answer

    \(\displaystyle ln(e^x+\sqrt{4+e^{2x}})+C\)

    Exercise \(\PageIndex{39}\)

    \(\displaystyle ∫\frac{cosx}{sin^2x+2sinx}dx\)

    Answer

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    Exercise \(\PageIndex{40}\)

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

    Answer

    \(\displaystyle lnx−\frac{1}{6}ln(x^6+1)−\frac{arctan(x^3)}{3x^3}+C\)

    Exercise \(\PageIndex{41}\)

    \(\displaystyle ∫\frac{ln|x|arcsin(ln|x|)}{x}dx\)

    Answer

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    Use tables to perform the integration.

    Exercise \(\PageIndex{42}\)

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

    Answer

    \(\displaystyle ln∣x+\sqrt{16+x^2}∣+C\)

    Exercise \(\PageIndex{43}\)

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

    Answer

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    Exercise \(\PageIndex{44}\)

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

    Answer

    \(\displaystyle −\frac{1}{4}cot(2x)+C\)

    Exercise \(\PageIndex{45}\)

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

    Answer

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    Exercise \(\PageIndex{46}\)

    Find the area bounded by \(\displaystyle y(4+25x^2)=5,x=0,y=0,\) and \(\displaystyle x=4.\) Use a table of integrals or a CAS.

    Answer

    \(\displaystyle \frac{1}{2}arctan10\)

    Exercise \(\PageIndex{47}\)

    The region bounded between the curve \(\displaystyle y=\frac{1}{\sqrt{1+cosx}}, 0.3≤x≤1.1,\) 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.)

    Answer

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    Exercise \(\PageIndex{48}\)

    Use substitution and a table of integrals to find the area of the surface generated by revolving the curve \(\displaystyle y=e^x,0≤x≤3,\) about the x-axis. (Round the answer to two decimal places.)

    Answer

    1276.14

    Exercise \(\PageIndex{49}\)

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

    Answer

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    Exercise \(\PageIndex{50}\)

    Use a CAS or tables to find the area of the surface generated by revolving the curve \(\displaystyle y=cosx,0≤x≤\frac{π}{2},\) about the x-axis. (Round the answer to two decimal places.)

    Answer

    7.21

    Exercise \(\PageIndex{51}\)

    Find the length of the curve \(\displaystyle y=\frac{x^2}{4}\) over \(\displaystyle [0,8]\).

    Answer

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    Exercise \(\PageIndex{52}\)

    Find the length of the curve \(\displaystyle y=e^x\) over \(\displaystyle [0,ln(2)].\)

    Answer

    \(\displaystyle \sqrt{5}−\sqrt{2}+ln∣\frac{2+2\sqrt{2}}{1+\sqrt{5}}∣\)

    Exercise \(\PageIndex{53}\)

    Find the area of the surface formed by revolving the graph of \(\displaystyle y=2\sqrt{x}\) over the interval \(\displaystyle [0,9]\) about the x-axis.

    Answer

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    Exercise \(\PageIndex{54}\)

    Find the average value of the function \(\displaystyle f(x)=\frac{1}{x^2+1}\) over the interval \(\displaystyle [−3,3].\)

    Answer

    \(\displaystyle \frac{1}{3}arctan(3)≈0.416\)

    Exercise \(\PageIndex{55}\)

    Approximate the arc length of the curve \(\displaystyle y=tan(πx)\) over the interval \(\displaystyle [0,\frac{1}{4}]\). (Round the answer to three decimal places.)

    Answer

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