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# 2.7E: Exercises

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## Other Strategies for Integration

Use a table of integrals to evaluate the following integrals.

Exercise $$\PageIndex{1}$$

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

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Exercise $$\PageIndex{2}$$

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

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

Exercise $$\PageIndex{3}$$

$$\displaystyle ∫x^3\sqrt{1+2x^2}dx$$

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Exercise $$\PageIndex{4}$$

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

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

Exercise $$\PageIndex{5}$$

$$\displaystyle ∫\frac{x}{x+1}dx$$

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Exercise $$\PageIndex{6}$$

$$\displaystyle ∫x⋅2^{x^2}dx$$

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

Exercise $$\PageIndex{7}$$

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

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Exercise $$\PageIndex{8}$$

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

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

Exercise $$\PageIndex{9}$$

$$\displaystyle ∫sin^3(2x)cos(2x)dx$$

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Exercise $$\PageIndex{10}$$

$$\displaystyle ∫csc(2w)cot(2w)dw$$

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

Exercise $$\PageIndex{11}$$

$$\displaystyle ∫2^ydy$$

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Exercise $$\PageIndex{12}$$

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

$$\displaystyle 9−6\sqrt{2}$$

Exercise $$\PageIndex{13}$$

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

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Exercise $$\PageIndex{14}$$

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

$$\displaystyle 2−\frac{π}{2}$$

Exercise $$\PageIndex{15}$$

$$\displaystyle ∫cos^3xdx$$

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Exercise $$\PageIndex{16}$$

$$\displaystyle ∫tan^5(3x)dx$$

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

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 $$\PageIndex{18}$$

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

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

Exercise $$\PageIndex{19}$$

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

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Exercise $$\PageIndex{20}$$

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

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

Exercise $$\PageIndex{21}$$

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

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Exercise $$\PageIndex{22}$$

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

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

Exercise $$\PageIndex{23}$$

$$\displaystyle ∫\frac{dx}{x\sqrt{x−1}}$$

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Exercise $$\PageIndex{24}$$

$$\displaystyle ∫x^3sinxdx$$

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

Exercise $$\PageIndex{25}$$

$$\displaystyle ∫x\sqrt{x^4−9}dx$$

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Exercise $$\PageIndex{26}$$

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

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

Exercise $$\PageIndex{27}$$

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

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Exercise $$\PageIndex{28}$$

$$\displaystyle ∫\frac{dx}{x\sqrt{x−1}}$$

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

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 $$\PageIndex{30}$$

$$\displaystyle ∫^{π/4}_0cos(2x)dx$$

$$\displaystyle 0.5=\frac{1}{2}$$

Exercise $$\PageIndex{31}$$

$$\displaystyle ∫^1_0x⋅e^{−x^2}dx$$

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Exercise $$\PageIndex{32}$$

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

$$\displaystyle 8.0$$

Exercise $$\PageIndex{33}$$

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

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Exercise $$\PageIndex{34}$$

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

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

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 $$\PageIndex{36}$$

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

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

Exercise $$\PageIndex{37}$$

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

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Exercise $$\PageIndex{38}$$

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

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

Exercise $$\PageIndex{39}$$

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

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Exercise $$\PageIndex{40}$$

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

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

Exercise $$\PageIndex{41}$$

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

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Use tables to perform the integration.

Exercise $$\PageIndex{42}$$

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

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

Exercise $$\PageIndex{43}$$

$$\displaystyle ∫\frac{3x}{2x+7}dx$$

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Exercise $$\PageIndex{44}$$

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

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

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.

$$\displaystyle \frac{1}{2}arctan10$$

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.)

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.)

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 $$\displaystyle [0,ln(2)].$$

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

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 $$\displaystyle [−3,3].$$

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