8.5E: Exercises for Section 8.5

Use a table of integrals to evaluate the following integrals.

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

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

Answer

$\displaystyle ∫\frac{x+3}{x^2+2x+2}\,dx = \tfrac{1}{2}\ln |x^2+2x+2|+2\arctan(x+1)+C$

3) $\displaystyle ∫x^3\sqrt{1+2x^2}\,dx$

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

Answer

$\displaystyle ∫\frac{1}{\sqrt{x^2+6x}}\,dx = \cosh^{−1}\left(\frac{x+3}{3}\right)+C$

5) $\displaystyle ∫\frac{x}{x+1}\,dx$

6) $\displaystyle ∫x⋅2^{x^2}\,dx$

Answer

$\displaystyle ∫x⋅2^{x^2}\,dx = \frac{2^{x^2−1}}{\ln 2}+C$

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

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

Answer

$\displaystyle ∫\frac{dy}{\sqrt{4−y^2}} = \arcsin\left(\frac{y}{2}\right)+C$

9) $\displaystyle ∫\sin^3(2x)\cos(2x)\,dx$

10) $\displaystyle ∫\csc(2w)\cot(2w)\,dw$

Answer

$\displaystyle ∫\csc(2w)\cot(2w)\,dw = −\tfrac{1}{2}\csc(2w)+C$

11) $\displaystyle ∫2^y\,dy$

12) $\displaystyle ∫^1_0\frac{3x}{\sqrt{x^2+8}}\,dx$

Answer

$\displaystyle ∫^1_0\frac{3x}{\sqrt{x^2+8}}\,dx = 9−6\sqrt{2}$

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

14) $\displaystyle ∫^{π/2}_0\tan^2\left(\frac{x}{2}\right)\,dx$

Answer

$\displaystyle ∫^{π/2}_0\tan^2\left(\frac{x}{2}\right)\,dx = 2−\frac{π}{2}$

15) $\displaystyle ∫\cos^3x\,dx$

16) $\displaystyle ∫\tan^5(3x)\,dx$

Answer

$\displaystyle ∫\tan^5(3x)\,dx = \tfrac{1}{12}\tan^4(3x)−\tfrac{1}{6}\tan^2(3x)+\tfrac{1}{3}\ln|\sec 3x|+C$

17) $\displaystyle ∫\sin^2y\cos^3y\,dy$

Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers.

18) [T] $\displaystyle ∫\frac{dw}{1+\sec\left(\frac{w}{2}\right)}$

Answer

$\displaystyle ∫\frac{dw}{1+\sec\left(\frac{w}{2}\right)} = 2\cot\left(\tfrac{w}{2}\right)−2\csc\left(\tfrac{w}{2}\right)+w+C$

19) [T] $\displaystyle ∫\frac{dw}{1−\cos(7w)}$

20) [T] $\displaystyle ∫^t_0\frac{dt}{4\cos t+3\sin t}$

Answer

$\displaystyle ∫^t_0\frac{dt}{4\cos t+3\sin t} = \tfrac{1}{5}\ln\Big|\frac{2(5+4\sin t−3\cos t)}{4\cos t+3\sin t}\Big|$

21) [T] $\displaystyle ∫\frac{\sqrt{x^2−9}}{3x}\,dx$

22) [T] $\displaystyle ∫\frac{dx}{x^{1/2}+x^{1/3}}$

Answer

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

23) [T] $\displaystyle ∫\frac{dx}{x\sqrt{x−1}}$

24) [T] $\displaystyle ∫x^3\sin x\,dx$

Answer

$\displaystyle ∫x^3\sin x\,dx = −x^3\cos x+3x^2\sin x+6x\cos x−6\sin x+C$

25) [T] $\displaystyle ∫x\sqrt{x^4−9}\,dx$

26) [T] $\displaystyle ∫\frac{x}{1+e^{−x^2}}\,dx$

Answer

$\displaystyle ∫\frac{x}{1+e^{−x^2}}\,dx = \tfrac{1}{2}\left(x^2+\ln|1+e^{−x^2}|\right)+C$

27) [T] $\displaystyle ∫\frac{\sqrt{3−5x}}{2x}\,dx$

28) [T] $\displaystyle ∫\frac{dx}{x\sqrt{x−1}}$

Answer

$\displaystyle ∫\frac{dx}{x\sqrt{x−1}} = 2\arctan\big(\sqrt{x−1}\big)+C$

29) [T] $\displaystyle ∫e^x\cos^{−1}(e^x)\,dx$

Use a calculator or CAS to evaluate the following integrals.

30) [T] $\displaystyle ∫^{π/4}_0\cos 2x \, dx$

Answer

$\displaystyle ∫^{π/4}_0\cos 2x \, dx = 0.5=\frac{1}{2}$

31) [T] $\displaystyle ∫^1_0x⋅e^{−x^2}\,dx$

32) [T] $\displaystyle ∫^8_0\frac{2x}{\sqrt{x^2+36}}\,dx$

Answer

$\displaystyle ∫^8_0\frac{2x}{\sqrt{x^2+36}}\,dx = 8.0$

33) [T] $\displaystyle ∫^{2/\sqrt{3}}_0\frac{1}{4+9x^2}\,dx$

34) [T] $\displaystyle ∫\frac{dx}{x^2+4x+13}$

Answer

$\displaystyle ∫\frac{dx}{x^2+4x+13} = \tfrac{1}{3}\arctan\left(\tfrac{1}{3}(x+2)\right)+C$

35) [T] $\displaystyle ∫\frac{dx}{1+\sin x}$

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.

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

Answer

$\displaystyle ∫\frac{dx}{x^2+2x+10} = \tfrac{1}{3}\arctan\left(\frac{x+1}{3}\right)+C$

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

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

Answer

$\displaystyle ∫\frac{e^x}{\sqrt{e^{2x}−4}}\,dx = \ln\left(e^x+\sqrt{4+e^{2x}}\right)+C$

39) $\displaystyle ∫\frac{\cos x}{\sin^2x+2\sin x}\,dx$

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

Answer

$\displaystyle ∫\frac{\arctan(x^3)}{x^4}\,dx = \ln x−\tfrac{1}{6}\ln(x^6+1)−\frac{\arctan(x^3)}{3x^3}+C$

41) $\displaystyle ∫\frac{\ln|x|\arcsin\left(\ln|x|\right)}{x}\,dx$

Use tables to perform the integration.

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

Answer

$\displaystyle ∫\frac{dx}{\sqrt{x^2+16}} = \ln |x|+\sqrt{16+x^2}∣+C$

43) $\displaystyle ∫\frac{3x}{2x+7}\,dx$

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

Answer

$\displaystyle ∫\frac{dx}{1−\cos 4x} = −\frac{1}{4}\cot 2x+C$

45) $\displaystyle ∫\frac{dx}{\sqrt{4x+1}}$

46) Find the area bounded by $y(4+25x^2)=5,\;x=0,\;y=0,$ and $x=4.$ Use a table of integrals or a CAS.

Answer

$\frac{1}{2}\arctan 10$ units²

47) The region bounded between the curve $y=\dfrac{1}{\sqrt{1+\cos x}}, \; 0.3≤x≤1.1,$ 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.)

48) Use substitution and a table of integrals to find the area of the surface generated by revolving the curve $y=e^x,\; 0≤x≤3,$ about the $x$-axis. (Round the answer to two decimal places.)

Answer

$1276.14$ units²

49) [T] Use an integral table and a calculator to find the area of the surface generated by revolving the curve $y=\dfrac{x^2}{2},\; 0≤x≤1,$ about the $x$-axis. (Round the answer to two decimal places.)

50) [T] Use a CAS or tables to find the area of the surface generated by revolving the curve $y=\cos x,\; 0≤x≤\frac{π}{2},$ about the $x$-axis. (Round the answer to two decimal places.)

Answer

$7.21$ units²

51) Find the length of the curve $y=\dfrac{x^2}{4}$ over $[0,8]$.

52) Find the length of the curve $y=e^x$ over $[0,\,\ln(2)].$

Answer

$\left(\sqrt{5}−\sqrt{2}+\ln\Big|\frac{2+2\sqrt{2}}{1+\sqrt{5}}\Big|\right)$ units

53) Find the area of the surface formed by revolving the graph of $y=2\sqrt{x}$ over the interval $[0,9]$ about the $x$-axis.

54) Find the average value of the function $f(x)=\dfrac{1}{x^2+1}$ over the interval $[−3,3].$

Answer

$\frac{1}{3}\arctan(3)≈0.416$

55) Approximate the arc length of the curve $y=\tan πx$ over the interval $\left[0,\frac{1}{4}\right]$. (Round the answer to three decimal places.)

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.