2.7E: Exercises
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( \newcommand{\kernel}{\mathrm{null}\,}\)
Other Strategies for Integration
Use a table of integrals to evaluate the following integrals.
Exercise 2.7E.1
∫40x√1+2xdx
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Exercise 2.7E.2
∫x+3x2+2x+2dx
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12ln∣x2+2x+2∣+2arctan(x+1)+C
Exercise 2.7E.3
∫x3√1+2x2dx
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Exercise 2.7E.4
∫1√x2+6xdx
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cosh−1(x+33)+C
Exercise 2.7E.5
∫xx+1dx
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Exercise 2.7E.6
∫x⋅2x2dx
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2x2−1ln2+C
Exercise 2.7E.7
∫14x2+25dx
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Exercise 2.7E.8
∫dy√4−y2
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arcsin(y2)+C
Exercise 2.7E.9
∫sin3(2x)cos(2x)dx
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Exercise 2.7E.10
∫csc(2w)cot(2w)dw
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−12csc(2w)+C
Exercise 2.7E.11
∫2ydy
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Exercise 2.7E.12
∫103xdx√x2+8
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9−6√2
Exercise 2.7E.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.
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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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