Practice Problems

Work through the odd-numbered problems 1-69. Once you have completed the problem set, check your answers.

Practice Problems

Answers

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}