$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 7.5E: Exercises for Section 7.5

• • OpenStax
• OpenStax
$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

Use a table of integrals to evaluate the following integrals.

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

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

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

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

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

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

5) $$\displaystyle ∫\frac{x}{x+1}\,dx$$

6) $$\displaystyle ∫x⋅2^{x^2}\,dx$$

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

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

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

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

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

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

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

11) $$\displaystyle ∫2^y\,dy$$

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

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

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

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

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

15) $$\displaystyle ∫\cos^3x\,dx$$

16) $$\displaystyle ∫\tan^5(3x)\,dx$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Use a calculator or CAS to evaluate the following integrals.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Use tables to perform the integration.

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

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

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

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

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

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.

$$\frac{1}{2}\arctan 10$$ 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.)

$$1276.14$$ 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.)

$$7.21$$ 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 $$[0,\,\ln(2)].$$

$$\left(\sqrt{5}−\sqrt{2}+\ln\Big|\frac{2+2\sqrt{2}}{1+\sqrt{5}}\Big|\right)$$ 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 $$[−3,3].$$

$$\frac{1}{3}\arctan(3)≈0.416$$

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.)