Practice Problems

For problems 1 - 3, put $\mathrm{f}(x)=x^{2}, \mathrm{~g}(x)=x$ and verify that

1. $\int_{1}^{2} \mathrm{f}(x) \cdot \mathrm{g}(x) \mathrm{d} \mathrm{x} \neq \int_{1}^{2} \mathrm{f}(x) \mathrm{d} \mathrm{x} \cdot \int_{1}^{2} \mathrm{~g}(x) \mathrm{d} \mathrm{x}$

3. $\begin{aligned} &\int_{0}^{1} \mathrm{f}(x) \cdot \mathrm{g}(x) \mathrm{d} x \neq \int_{0}^{1} \mathrm{f}(x) \mathrm{d} \mathrm{x} \cdot \int_{0}^{1} \mathrm{~g}(x) \mathrm{dx} \\ &0 \end{aligned}$

For problems 5 – 13 , use the suggested $u$ to find du and rewrite the integral in terms of $u$ and $du$. Then find an antiderivative in terms of $u$ , and, finally, rewrite your answer in terms of $x$.

5. $\int \cos (3 x) \mathrm{d} x \quad u=3 x$

7. $\int e^{x} \sin \left(2+e^{x}\right) d x \quad u=2+e^{x}$

9. $\int \cos (x) \sec ^{2}(\sin (x)) \mathrm{dx} u=\sin (x)$

11. $\int \frac{5}{3+2 x} \mathrm{~d} x \quad u=3+2 x$

13. $\int x^{2} \sin \left(1+x^{3}\right) \mathrm{dx} \quad u=1+x^{3}$

For problems 15 – 25 , use the change of variable technique to find an antiderivative in terms of $x$ .

15. $\int \cos (4 x) \mathrm{dx}$

17. $\int x^{3}\left(5+x^{4}\right)^{11} \mathrm{dx}$

19. $\int \frac{3 x^{2}}{2+x^{3}} \mathrm{dx}$

21. $\int \frac{\ln (x)}{x} \mathrm{dx}$

23. $\int(1+3 x)^{7} \mathrm{dx}$

25. $\int e^{x} \cdot \sec \left(e^{x}\right) \cdot \tan \left(e^{x}\right) d x$

For problems 27 – 37 , evaluate the definite integrals.

27. $\int_{0}^{\pi / 2} \cos (3 x) \mathrm{dx}$

29. $\int_{0}^{1} \mathrm{e}^{x} \cdot \sin \left(2+e^{x}\right) \mathrm{d} x$

31. $\int_{-1}^{1} x^{2}\left(1+x^{3}\right)^{5} \mathrm{~d} \mathrm{x}$

33. $\int_{0}^{2} \frac{5}{3+2 x} d x$

35. $\int_{0}^{1} x \sqrt{1-x^{2}} \mathrm{dx}$

37. $\begin{aligned} &\int_{0}^{1} \sqrt{1+3 x} \mathrm{dx} \end{aligned}$

39. $\int \sin ^{2}(5 x) d x$

41. $\int \frac{1}{2}-\sin ^{2}(x) \mathrm{dx}$

43. Find the area under one arch of the $y=\sin ^{2}(x)$ graph.

Problems 45 – 53 , expand the integrand and then find an antiderivative.

45. $\int\left(x^{2}+1\right)^{3} \mathrm{dx}$

47. $\int\left(e^{x}+1\right)^{2} d x$

49. $\int\left(x^{2}+1\right)\left(x^{3}+5\right) \mathrm{dx}$

51. $\int \mathrm{e}^{x}\left(\mathrm{e}^{x}+\mathrm{e}^{3 x}\right) \mathrm{dx}$

53. $\int \sqrt{x}\left(x^{2}+3 x-2\right) d x$

Problems 53 – 63 , perform the division and then find an antiderivative.

55. $\int \frac{3 x}{x+1} \mathrm{dx} \quad\left(\frac{3 x}{x+1}=3-\frac{3}{x+1}\right)$

57. $\int \frac{x^{2}-1}{x+1} d x$

59. $\int \frac{2 x^{2}-13 x+18}{x-1} \mathrm{dx}$

61. $\int \frac{x+2}{x-1} d x$

63. $\int \frac{x+4}{\sqrt{x}} \mathrm{dx}$

The definite integrals in problems 65 – 69 involve areas associated with parts of circles (Fig. 2). Use your knowledge of circles and their areas to evaluate the integrals. (Suggestion: Sketch a graph of the integrand function.)

65. $\int_{-1}^{1} \sqrt{1-x^{2}} d x$

67. $\int_{-3}^{3} \sqrt{9-x^{2}} d x$

69. $\int_{-1}^{1} 2+\sqrt{1-x^{2}} d x$

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-5.7-Finding-Antiderivatives.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.

1. Left side = $\left.\frac{1}{4} x^{4}\right|_{1} ^{2}=\frac{15}{4}$. Right side = $\left\{\left.\frac{1}{3} \mathrm{x}^{3}\right|_{1} ^{2}=\frac{7}{3}\right\} \cdot\left\{\left.\frac{1}{2} \mathrm{x}^{2}\right|_{1} ^{2}=\frac{3}{2}\right\}=\frac{7}{2} \neq$ left side.

3. Left side = $\frac{1}{4}$. Right side = $\left(\frac{1}{3}\right) \cdot\left(\frac{1}{2}\right)=\frac{1}{6} \neq$ left side.

5. $\frac{1}{3} \sin (3 x)+\mathrm{C}$

7. $-\cos \left(2+\mathrm{e}^{\mathrm{x}}\right)+\mathrm{C}$

9. $\tan (\sin (x))+C$

11. $\frac{5}{2} \ln |3+2 \mathrm{x}|+\mathrm{C}$

13. $-\frac{1}{3} \cos \left(1+x^{3}\right)+C$

15. $\frac{1}{4} \sin (4 x)+C$

17. $\frac{1}{48}\left(5+x^{4}\right)^{12}+C$

19. $\ln \left|2+x^{3}\right|+C$

21. $\frac{1}{2}(\ln (x))^{2}+C$

23. $\frac{1}{24}(1+3 x)^{8}+C$

25. $\sec \left(\mathrm{e}^{\mathrm{x}}\right)+\mathrm{C}$

27. $\left.\frac{1}{3} \sin (3 x)\right|_{0} ^{\pi / 2}=-\frac{1}{3}$

29. $-\left.\cos \left(2+\mathrm{e}^{\mathrm{x}}\right)\right|_{0} ^{1}=\cos (3)-\cos (2+\mathrm{e}) \approx-0.996$

31. $\left.\frac{1}{18}\left(1+x^{3}\right)^{6}\right|_{-1} ^{1}=\frac{32}{9}$

33. $\left.\frac{5}{2} \ln |3+2 \mathrm{x}|\right|_{0} ^{2}=\frac{5}{2} \ln \left(\frac{7}{3}\right)$

35. $-\left.\frac{1}{3}\left(1-\mathrm{x}^{2}\right)^{3 / 2}\right|_{0} ^{1}=\frac{1}{3}$

37. $\left.\frac{2}{9}(1+3 x)^{3 / 2}\right|_{0} ^{1}=\frac{16}{9}-\frac{2}{9}=\frac{14}{9}$

39. $\frac{1}{2} x-\frac{1}{20} \sin (10 x)+C$

41. $\frac{1}{4} \sin (2 x)+C$

43. $\frac{1}{2} x-\left.\frac{1}{4} \sin (2 x)\right|_{0} ^{\pi}=\frac{\pi}{2}$

45. $\frac{1}{7} x^{7}+\frac{3}{5} x^{5}+x^{3}+x+C$

47. $\frac{1}{2} e^{2 x}+2 e^{x}+x+C$

49. $\frac{1}{6} x^{6}+\frac{1}{4} x^{4}+\frac{5}{3} x^{3}+5 x+C$

51. $\frac{1}{2} e^{2 x}+\frac{1}{4} e^{4 x}+C$

53. $\frac{2}{7} x^{7 / 2}+\frac{6}{5} x^{5 / 2}-\frac{4}{3} x^{3 / 2}+C$

55. $3 x-3 \cdot \ln |x+1|+C$

57. $\frac{1}{2} x^{2}-x+C$

59. (divide first) $x^{2}-11 x+7 \cdot \ln |x-1|+C$

61. (divide first) $x+3 \cdot \ln |x-1|+C$

63. $\frac{2}{3} x^{3 / 2}+8 x^{1 / 2}+C$

65. (area of semicirle with radius 1) = $\frac{1}{2} \pi(1)^{2}=\frac{\pi}{2}$

67. (area of semicirle with radius 3) = $\frac{1}{2} \pi(3)^{2}=\frac{9}{2} \pi$

69. (area of rectangle) + (area of semicircle of radius 1) = $(2)(2)+\frac{1}{2}\left(\pi(1)^{2}\right)=4+\frac{\pi}{2}$