Loading [MathJax]/jax/element/mml/optable/GreekAndCoptic.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

2.7E: Exercises

This page is a draft and is under active development. 

( \newcommand{\kernel}{\mathrm{null}\,}\)

Other Strategies for Integration

Use a table of integrals to evaluate the following integrals.

Exercise 2.7E.1

40x1+2xdx

Answer

Add texts here. Do not delete this text first.

Exercise 2.7E.2

x+3x2+2x+2dx

Answer

12lnx2+2x+2+2arctan(x+1)+C

Exercise 2.7E.3

x31+2x2dx

Answer

Add texts here. Do not delete this text first.

Exercise 2.7E.4

1x2+6xdx

Answer

cosh1(x+33)+C

Exercise 2.7E.5

xx+1dx

Answer

Add texts here. Do not delete this text first.

Exercise 2.7E.6

x2x2dx

Answer

2x21ln2+C

Exercise 2.7E.7

14x2+25dx

Answer

Add texts here. Do not delete this text first.

Exercise 2.7E.8

dy4y2

Answer

arcsin(y2)+C

Exercise 2.7E.9

sin3(2x)cos(2x)dx

Answer

Add texts here. Do not delete this text first.

Exercise 2.7E.10

csc(2w)cot(2w)dw

Answer

12csc(2w)+C

Exercise 2.7E.11

2ydy

Answer

Add texts here. Do not delete this text first.

Exercise 2.7E.12

103xdxx2+8

Answer

962

Exercise 2.7E.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.


2.7E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

Support Center

How can we help?