## Practice Problems

Work through the odd-numbered problems 1-67. Once you have completed the problem set, check your answers.

### Practice Problems

1. $\mathrm{A}(\mathrm{x})=\int_{0}^{x} 3 t^{2} \mathrm{dt}$

(a) Use part 2 of the Fundamental Theorem to find a formula for $\mathrm{A}(x)$ and then differentiate $\mathrm{A}(x)$ to obtain a formula for $\mathrm{A}^{\prime}(x)$. Evaluate $\mathrm{A}^{\prime}(x)$ at $x=1,2$ and $3$.

(b) Use part 1 of the Fundamental Theorem to evaluate $\mathrm{A}^{\prime}(x)$ at $x=1,2,$ and $3$.

In problems 3 – 7 , evaluate $\mathrm{A}^{\prime}(x)$ at $x=1,2,$ and $3$.

3. $\mathrm{A}(\mathrm{x})=\int_{0}^{x} 2 t \mathrm{dt}$

5. $\mathrm{A}(\mathrm{x})=\int_{-3}^{x} 2 t \mathrm{dt}$

7. $\mathrm{A}(\mathrm{x})=\int_{0}^{x} \sin (t) \mathrm{dt}$

In problems 9 – 11 $\mathrm{A}(x)=\int_{0}^{x} \mathrm{f}(t)$ for the functions in Figures 10 – 14. Evaluate $\mathrm{A}^{\prime}(1), \mathrm{A}^{\prime}(2), \mathrm{A}^{\prime}(3)$.

9. $f$ in Fig. 10

11. $f$ in Fig. 12

In problems 13 – 33, verify that $\mathrm{F}(x)$ is an antiderivative of the integrand $\mathrm{f}(x)$ and use Part 2 of the Fundamental Theorem to evaluate the definite integrals.

13. $\int_{0}^{1} 2 x \mathrm{dx}, \mathrm{F}(x)=x^{2}+5$

15. $\int_{1}^{3} x^{2} \mathrm{dx}, \mathrm{F}(x)=\frac{1}{3} x^{3}$

17. $\int_{1}^{5} \frac{1}{x} \mathrm{dx}, \mathrm{F}(x)=\ln (x)$

19. $\int_{1 / 2}^{3} \frac{1}{x} \mathrm{dx}, \mathrm{F}(x)=\ln (x)$

21. $\int_{0}^{\pi / 2} \cos (x) \mathrm{dx}, \mathrm{F}(x)=\sin (x)$

23. $\int_{0}^{1} \sqrt{x} \mathrm{dx}, \mathrm{F}(x)=\frac{2}{3} x^{3 / 2}$

25. $\int_{1}^{7} \sqrt{x} \mathrm{dx}, \mathrm{F}(x)=\frac{2}{3} x^{3 / 2}$

27. $\int_{1}^{9} \frac{1}{2 \sqrt{x}} \mathrm{dx}, \mathrm{F}(x)=\sqrt{x}$

29. $\int_{-2}^{3} \mathrm{e}^{x} \mathrm{dx}, \mathrm{F}(x)=\mathrm{e}^{x}$

31. $\int_{0}^{\pi / 4} \sec ^{2}(x) \mathrm{dx}, \mathrm{F}(\mathrm{x})=\tan (x)$

33. $\int_{0}^{3} 2 x \sqrt{1+x^{2}} \mathrm{dx}, \mathrm{F}(\mathrm{x})=\frac{2}{3}\left(1+x^{2}\right)^{3 / 2}$

For problems 33 – 47, find an antiderivative of the integrand and use Part 2 of the Fundamental Theorem to evaluate the definite integral.

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

37. $\int_{1}^{\mathrm{e}} \frac{1}{x} \mathrm{dx}$

39. $\int_{25}^{100} \sqrt{x} \mathrm{dx}$

41. $\int_{1}^{10} \frac{1}{x^{2}} d x$

43. $\int_{0}^{1} e^{x} d x$

45. $\int_{\pi / 6}^{\pi / 4} \sec ^{2}(x) d x$

47. $\int_{3}^{3} \sin (x) \cdot \ln (x) d x$

In problems 49 – 53 , find the area of each shaded region.

49. Region in Fig. 14.

51. Region in Fig. 16.

53. Region in Fig. 18.

Leibniz' Rule

55. If $\mathbf{D}(A(x))=\tan (x)$, then find $\mathbf{D}(\mathrm{A}(3 x)), \mathbf{D}\left(\mathrm{A}\left(x^{2}\right)\right)$ and $\mathbf{D}(\mathrm{A}(\sin (x)))$.

57. $\frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{1}^{5 x} \sqrt{1+t} \mathrm{dt}\right)$

59. $\frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{0}^{\sin (x)} \sqrt{1+t} \mathrm{dt}\right.$

61. $\frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{0}^{1-2 x} 3 t^{2}+2 \mathrm{dt}\right)$

63. $\frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{x}^{\pi} \cos (3 t) \mathrm{dt}\right)$

65. $\frac{\mathrm{d}}{\mathrm{d} \mathrm{x}}\left(\int_{x}^{x^{2}} \tan (t) \mathrm{dt}\right)$

67. $\frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{2}^{\ln (x)} 5 t \cos (3 t) \mathrm{dt}\right)$

Source: Dale Hoffman, https://learn.saylor.org/pluginfile.php/1403575/mod_resource/content/2/CC_4_5_FundamentalThm.pdf