2.7E: Exercises

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Dec 21, 2020
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- 2.7: Other Strategies for Integration
- 2.E: Chapter Review Exercises

This page is a draft and is under active development.

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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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

Exercise $\PageIndex{10}$

$\displaystyle ∫csc(2w)cot(2w)dw$

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

Exercise $\PageIndex{11}$

$\displaystyle ∫2^ydy$

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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