Practice Problems

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
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