MA005: Calculus I

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
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