Practice Problems

Work through the odd-numbered problems 1-55. Once you have completed the problem set, check your answers.

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,
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