Practice Problems

Site: Saylor Academy
Course: MA005: Calculus I
Book: Practice Problems
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Date: Saturday, July 13, 2024, 5:07 AM

Description

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

Table of contents

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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Answers

1. Table #35, a=2: \quad \frac{1}{2} \arctan \left(\frac{x}{2}\right)+C

3. Table #35, \mathrm{a}=5: \quad \mathrm{x}^{2}+\frac{2}{5} \arctan \left(\frac{\mathrm{x}}{5}\right)+\mathrm{C}

5. Table #37, \mathrm{a}=3: \quad\left(\frac{1}{3}\right) \ln \mid \frac{x+3}{x-3} 1+C

7. Table #35, \mathrm{a}=\sqrt{3}: \frac{1}{\sqrt{3}} \arctan \left(\frac{\mathrm{x}}{\sqrt{3}}\right)+\mathrm{C}

9. Table #35, a=\sqrt{2}: e^{x}+\frac{7}{\sqrt{2}} \arctan \left(\frac{x}{\sqrt{2}}\right)+C

11. Table #34, a=\sqrt{5}: 3 \cdot \arcsin \left(\frac{x}{\sqrt{5}}\right)+C

13. Table #35, \mathrm{a}=\frac{2}{5}: \frac{1}{10} \arctan \left(\frac{5}{2} \mathrm{x}\right)+\mathrm{C}

15. First substitute \mathbf{u}=2 \mathrm{x}, \mathrm{d} \mathrm{u}=2 \mathrm{dx}. Then use Table #34 with \mathrm{a}=1: \frac{5}{2} \cdot \arcsin (\mathrm{u})+\mathrm{C}=\frac{5}{2} \cdot \arcsin (2 \mathrm{x})+\mathrm{C}

17. Table #43 and substitution \mathrm{u}=3 \mathrm{x}: \frac{2}{3} \ln \mid 3 \mathrm{x}+\sqrt{1+9 \mathrm{x}^{2}} \mathrm{I}+\mathrm{C}.

19. Table #38 and substitution \mathrm{u}=\mathrm{x}+1:(\mathrm{x}+1) \cdot \ln \mid \mathrm{x}+11-(\mathrm{x}+1)+\mathrm{C} \text { or }(\mathrm{x}+1) \cdot \ln \mid \mathrm{x}+1 \mathrm{l}-\mathrm{x}+\mathrm{C}_{2}

21. Table #38 and substitution u=5 x^{2}+7: \frac{3}{10}\left\{\left(5 x^{2}+7\right) \cdot \ln 15 x^{2}+7 \mid-\left(5 x^{2}+7\right)\right\}+C

23. Table #38 and substitution \mathrm{u}=\sin (\mathrm{x}): \sin (\mathrm{x}) \cdot \ln |\sin (\mathrm{x})|-\sin (\mathrm{x})+\mathrm{C}

25. Table #44, \mathrm{a}=2: \frac{\mathrm{x}}{2} \sqrt{\mathrm{x}^{2}+4}+\frac{1}{2}(4) \cdot \ln \mid \mathrm{x}+\sqrt{\mathrm{x}^{2}+4} \mathrm{I}+\mathrm{C}

27. Table #44, a=4: \frac{x}{2} \sqrt{x^{2}+16}+\frac{1}{2}(16) \cdot \ln \mid x+\sqrt{x^{2}+16} 1+C

29. Table #35, \mathrm{a}=5: \mathrm{x}^{2}+\left.\frac{2}{5} \arctan \left(\frac{\mathrm{x}}{5}\right)\right|_{1} ^{3}=8+\frac{2}{5}\left\{\arctan \left(\frac{3}{5}\right)-\operatorname{acrtan}\left(\frac{1}{5}\right)\right\}

31. Table #35, \mathrm{a}=\sqrt{3}:\left.\frac{1}{\sqrt{3}} \operatorname{acrtan}\left(\frac{\mathrm{x}}{\sqrt{3}}\right)\right|_{-1} ^{1}=\frac{1}{\sqrt{3}}\left\{\arctan \left(\frac{1}{\sqrt{3}}\right)-\arctan \left(\frac{-1}{\sqrt{3}}\right)\right\}

33. Table #34, \mathrm{a}=\sqrt{5}:\left.3 \arcsin \left(\frac{\mathrm{x}}{\sqrt{5}}\right)\right|_{1} ^{2}=3 \cdot\left\{\operatorname { a r c s i n } \left(\frac{2}{\sqrt{5}}-\arcsin \left(\frac{1}{\sqrt{5}}\right\}\right.\right.

35. Table #34, a=1 / 2:\left.\frac{5}{2} \arcsin \left(\frac{x}{1 / 2}\right)\right|_{0} ^{0.1}=\frac{5}{2} \arcsin (0.2)

37. 7 \cdot \ln (7)-6

39. 3 \cdot \ln (3)-2 \cdot \ln (2)-1

41. 3 \sqrt{18}+\frac{9}{2} \cdot \ln \left(\frac{3+\sqrt{18}}{-3+\sqrt{18}}\right)


43. Table #19a: \frac{-\sin ^{2}(x) \cdot \cos (x)}{3}-\frac{2}{3} \cos (x)+C

45. Table #20: \frac{\cos ^{4}(\mathrm{x}) \cdot \sin (\mathrm{x})}{5}+\frac{4}{5} { answer to number 44 }

47. Table #29: x^{2} \cdot \sin (x)+2 x^{*} \cos (x)-2 \cdot \sin (x)+C


49. Average of \sin (x) on [0, \pi] is  \frac{2}{\pi}. Average of \sin ^{2}(x) on [0, \pi] is \frac{1}{2} \cdot \frac{2}{\pi}>\frac{1}{2}

51. Using results from #50: (a) \frac{1}{\mathrm{e}-1}                                           (b) \frac{1}{9}\{10 \cdot \ln (10)-9\} \approx \frac{14.03}{9} \approx 1.56 

(c) \frac{1}{99}\{100 \cdot \ln (100)-99\} \approx 361.52 / 99 \approx 3.65                (d) \frac{1}{199}\{200 \cdot \ln (200)-199\} \approx 860.66 / 199=4.32

53. (c) is largest.

55. approximately (a) 1.57 (b) 2.94 (c) 3.04 (d) 3.07 (e) 3.09