2.7E: Exercises
- Page ID
- 18589
This page is a draft and is under active development.
Other Strategies for Integration
Use a table of integrals to evaluate the following integrals.
Exercise \(\PageIndex{1}\)
\(\displaystyle ∫_0^4\frac{x}{\sqrt{1+2x}}dx\)
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Exercise \(\PageIndex{2}\)
\(\displaystyle ∫\frac{x+3}{x^2+2x+2}dx\)
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\(\displaystyle \frac{1}{2}ln∣x^2+2x+2∣+2arctan(x+1)+C\)
Exercise \(\PageIndex{3}\)
\(\displaystyle ∫x^3\sqrt{1+2x^2}dx\)
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Exercise \(\PageIndex{4}\)
\(\displaystyle ∫\frac{1}{\sqrt{x^2+6x}}dx\)
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\(\displaystyle cosh^{−1}(\frac{x+3}{3})+C\)
Exercise \(\PageIndex{5}\)
\(\displaystyle ∫\frac{x}{x+1}dx\)
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Exercise \(\PageIndex{6}\)
\(\displaystyle ∫x⋅2^{x^2}dx\)
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\(\displaystyle \frac{2^{x^2−1}}{ln2}+C\)
Exercise \(\PageIndex{7}\)
\(\displaystyle ∫\frac{1}{4x^2+25}dx\)
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Exercise \(\PageIndex{8}\)
\(\displaystyle ∫\frac{dy}{\sqrt{4−y^2}}\)
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\(\displaystyle arcsin(\frac{y}{2})+C\)
Exercise \(\PageIndex{9}\)
\(\displaystyle ∫sin^3(2x)cos(2x)dx\)
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Exercise \(\PageIndex{10}\)
\(\displaystyle ∫csc(2w)cot(2w)dw\)
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\(\displaystyle −\frac{1}{2}csc(2w)+C\)
Exercise \(\PageIndex{11}\)
\(\displaystyle ∫2^ydy\)
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Exercise \(\PageIndex{12}\)
\(\displaystyle ∫^1_0\frac{3xdx}{\sqrt{x^2+8}}\)
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\(\displaystyle 9−6\sqrt{2}\)
Exercise \(\PageIndex{13}\)
\(\displaystyle ∫^{1/4}_{−1/4}sec^2(πx)tan(πx)dx\)
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Exercise \(\PageIndex{14}\)
\(\displaystyle ∫^{π/2}_0tan^2(\frac{x}{2})dx\)
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\(\displaystyle 2−\frac{π}{2}\)
Exercise \(\PageIndex{15}\)
\(\displaystyle ∫cos^3xdx\)
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Exercise \(\PageIndex{16}\)
\(\displaystyle ∫tan^5(3x)dx\)
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\(\displaystyle \frac{1}{12}tan^4(3x)−\frac{1}{6}tan^2(3x)+\frac{1}{3}ln|sec(3x)|+C\)
Exercise \(\PageIndex{17}\)
\(\displaystyle ∫sin^2ycos^3ydy\)
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Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers.
Exercise \(\PageIndex{18}\)
\(\displaystyle ∫\frac{dw}{1+sec(\frac{w}{2})}\)
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\(\displaystyle 2cot(\frac{w}{2})−2csc(\frac{w}{2})+w+C\)
Exercise \(\PageIndex{19}\)
\(\displaystyle ∫\frac{dw}{1−cos(7w)}\)
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Exercise \(\PageIndex{20}\)
\(\displaystyle ∫^t_0\frac{dt}{4cost+3sint}\)
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\(\displaystyle \frac{1}{5}ln∣\frac{2(5+4sint−3cost)}{4cost+3sint}∣\)
Exercise \(\PageIndex{21}\)
\(\displaystyle ∫\frac{\sqrt{x^2−9}}{3x}dx\)
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Exercise \(\PageIndex{22}\)
\(\displaystyle ∫\frac{dx}{x^{1/2}+x^{1/3}}\)
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\(\displaystyle 6x^{1/6}−3x^{1/3}+2\sqrt{x}−6ln[1+x^{1/6}]+C\)
Exercise \(\PageIndex{23}\)
\(\displaystyle ∫\frac{dx}{x\sqrt{x−1}}\)
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Exercise \(\PageIndex{24}\)
\(\displaystyle ∫x^3sinxdx\)
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\(\displaystyle −x^3cosx+3x^2sinx+6xcosx−6sinx+C\)
Exercise \(\PageIndex{25}\)
\(\displaystyle ∫x\sqrt{x^4−9}dx\)
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Exercise \(\PageIndex{26}\)
\(\displaystyle ∫\frac{x}{1+e^{−x^2}}dx\)
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\(\displaystyle \frac{1}{2}(x^2+ln∣1+e^{−x^2}∣)+C\)
Exercise \(\PageIndex{27}\)
\(\displaystyle ∫\frac{\sqrt{3−5x}}{2x}dx\)
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Exercise \(\PageIndex{28}\)
\(\displaystyle ∫\frac{dx}{x\sqrt{x−1}}\)
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\(\displaystyle 2arctan(\sqrt{x−1})+C\)
Exercise \(\PageIndex{29}\)
\(\displaystyle ∫e^xcos^{−1}(e^x)dx\)
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Use a calculator or CAS to evaluate the following integrals.
Exercise \(\PageIndex{30}\)
\(\displaystyle ∫^{π/4}_0cos(2x)dx\)
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\(\displaystyle 0.5=\frac{1}{2}\)
Exercise \(\PageIndex{31}\)
\(\displaystyle ∫^1_0x⋅e^{−x^2}dx\)
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Exercise \(\PageIndex{32}\)
\(\displaystyle ∫^8_0\frac{2x}{\sqrt{x^2+36}}dx\)
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\(\displaystyle 8.0\)
Exercise \(\PageIndex{33}\)
\(\displaystyle ∫^{2/\sqrt{3}}_0\frac{1}{4+9x^2}dx\)
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Exercise \(\PageIndex{34}\)
\(\displaystyle ∫\frac{dx}{x^2+4x+13}\)
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\(\displaystyle \frac{1}{3}arctan(\frac{1}{3}(x+2))+C\)
Exercise \(\PageIndex{35}\)
\(\displaystyle ∫\frac{dx}{1+sinx}\)
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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 \(\PageIndex{36}\)
\(\displaystyle ∫\frac{dx}{x^2+2x+10}\)
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\(\displaystyle \frac{1}{3}arctan(\frac{x+1}{3})+C\)
Exercise \(\PageIndex{37}\)
\(\displaystyle ∫\frac{dx}{\sqrt{x^2−6x}}\)
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Exercise \(\PageIndex{38}\)
\(\displaystyle ∫\frac{e^x}{\sqrt{e^{2x}−4}}dx\)
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\(\displaystyle ln(e^x+\sqrt{4+e^{2x}})+C\)
Exercise \(\PageIndex{39}\)
\(\displaystyle ∫\frac{cosx}{sin^2x+2sinx}dx\)
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Exercise \(\PageIndex{40}\)
\(\displaystyle ∫\frac{arctan(x^3)}{x^4}dx\)
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\(\displaystyle lnx−\frac{1}{6}ln(x^6+1)−\frac{arctan(x^3)}{3x^3}+C\)
Exercise \(\PageIndex{41}\)
\(\displaystyle ∫\frac{ln|x|arcsin(ln|x|)}{x}dx\)
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Use tables to perform the integration.
Exercise \(\PageIndex{42}\)
\(\displaystyle ∫\frac{dx}{\sqrt{x^2+16}}\)
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\(\displaystyle ln∣x+\sqrt{16+x^2}∣+C\)
Exercise \(\PageIndex{43}\)
\(\displaystyle ∫\frac{3x}{2x+7}dx\)
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Exercise \(\PageIndex{44}\)
\(\displaystyle ∫\frac{dx}{1−cos(4x)}\)
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\(\displaystyle −\frac{1}{4}cot(2x)+C\)
Exercise \(\PageIndex{45}\)
\(\displaystyle ∫\frac{dx}{\sqrt{4x+1}}\)
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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
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\(\displaystyle \frac{1}{2}arctan10\)
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.)
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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.)
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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.)
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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.)
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7.21
Exercise \(\PageIndex{51}\)
Find the length of the curve \(\displaystyle y=\frac{x^2}{4}\) over \(\displaystyle [0,8]\).
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Exercise \(\PageIndex{52}\)
Find the length of the curve \(\displaystyle y=e^x\) over \(\displaystyle [0,ln(2)].\)
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\(\displaystyle \sqrt{5}−\sqrt{2}+ln∣\frac{2+2\sqrt{2}}{1+\sqrt{5}}∣\)
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.
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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 \(\displaystyle [−3,3].\)
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\(\displaystyle \frac{1}{3}arctan(3)≈0.416\)
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.)
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