Practice Problems

Use the integral table for the following problems.

1. $\int \frac{1}{4+x^{2}} \mathrm{dx}$

3. $\int 2 x+\frac{2}{25+x^{2}} d x$

5. $\int \frac{2}{9-x^{2}} d x$

7. $\int \frac{1}{3+x^{2}} \mathrm{dx}$

9. $\int e^{x}+\frac{7}{2+x^{2}} d x$

11. $\int \frac{3}{\sqrt{5-x^{2}}} \mathrm{dx}$

13. $\int \frac{1}{4+25 x^{2}} \mathrm{dx}$

15. $\int \frac{5}{\sqrt{1-4 x^{2}}} d x$

17. $\int \frac{2}{\sqrt{1+9 x^{2}}} \mathrm{dx}$

19. $\int \ln (x+1) \mathrm{dx}$

21. $\int 3 x \cdot \ln \left(5 x^{2}+7\right) \mathrm{dx}$

23. $\int \cos (x) \ln (\sin (x)) \mathrm{dx}$

25. $\int \sqrt{4+x^{2}} d x$

27. $\int \sqrt{16+x^{2}} d x$

29. $\int_{1}^{3} 2 x+\frac{2}{25+x^{2}} \mathrm{dx}$

31. $\int_{-1}^{1} \frac{1}{3+x^{2}} d x$

33. $\int_{1}^{2} \frac{3}{\sqrt{5-x^{2}}} \mathrm{dx}$

35. $\int_{0}^{0.1} \frac{5}{\sqrt{1-4 x^{2}}} d x$

37. $\int_{0}^{6} \ln (x+1) \mathrm{dx}$

39. $\int_{0}^{\pi / 2} \cos (x) \ln (2+\sin (x)) \mathrm{dx}$

41. $\int_{-3}^{3} \sqrt{9+x^{2}} d x$

In problems 43 – 47 , use the recursion formulas in the table

43. $\int \sin ^{3}(x) \mathrm{dx}$

45. $\int \cos ^{5}(x) \mathrm{dx}$

47. $\int x^{2} \cos (x) \mathrm{dx}$

49. Before doing any calculations, predict which do you expect to be larger; the average value of $\sin (x)$ or of $\sin ^{2}(x)$ on the interval $[0, \pi]$? Then calculate each average to see if your prediction was correct.

51. Find the average value of $\mathrm{f}(x)=\ln (x)$ on the interval $1 \leq x \leq C$ when $C=e, 10,100,200$

53. Before doing any calculations, predict which of the following integrals you expect to be the largest?

(a) $\int_{1}^{2} e^{x} d x$ (b) $\int_{1}^{2} x \mathrm{e}^{x} \mathrm{dx}$ (c) $\int_{1}^{2} x^{2} e^{x} d x$ Then calculate the value of each integral.

55. Evaluate $\int_{0}^{\mathrm{C}} \frac{2}{1+x^{2}} \mathrm{dx}$ fo $\mathrm{C}=1,10,20, \text { and } 30$. Before doing the calculation, estimate the value of the integral when $C = 40$.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-5.9-Using-Tables-to-Find-Antiderivatives.pdf
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