Practice Problems

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) = $(2)(2)+\frac{1}{2}\left(\pi(1)^{2}\right)=4+\frac{\pi}{2}$