# BC Calculus Mixed Integration Practice

• Class: 5H: BC Calculus
• Author: Peter Atlas
• Text: Calculus Finney, Demana, Waits, Kennedy

The problems in this worksheet may or may not involve techniques learned in chapter 8. Calculator Inactive.

1. $$\displaystyle \int sin^{-1}(x) \, dx =$$
1. $$\displaystyle \left( 1 - x^2 \right) ^{-\frac{1}{2}} + C$$
2. $$\displaystyle x\sin^{-1}{x} + \left( 1 - x^2 \right) ^{\frac{1}{2}} + C$$
3. $$\displaystyle x\sin^{-1}{x} + \frac{1}{2} \left( 1 - x^2 \right) ^{\frac{1}{2}} + C$$
4. $$\displaystyle x\sin^{-1}{x} - \left( 1 - x^2 \right) ^{\frac{1}{2}} + C$$
5. $$\displaystyle x\sin^{-1}{x} + \frac{1}{2} \ln { \left| 1 - x^2 \right|} + C$$
2. Solution B
3. $$\displaystyle \int_2^3 \frac{dx}{x^2 - 1} =$$
1. $$\displaystyle \frac{1}{2} \ln { \frac{2}{3}}$$
2. $$\displaystyle \frac{1}{2} \ln { \frac{3}{2} }$$
3. $$\displaystyle \ln { \frac{3}{2}}$$
4. $$\displaystyle \ln { \frac{2}{3} }$$
5. $$\displaystyle 2 \ln { \frac{3}{2}}$$
4. Solution B
5. Find the area bounded by $$\displaystyle f(x) = \frac{1}{x(2 - x)}$$ and the x-axis on the interval [3, 5].
1. $$\displaystyle \frac{1}{2} \ln { \frac{5}{9}}$$
2. $$\displaystyle \ln { \frac{5}{9}}$$
3. $$\displaystyle \ln { \frac{3}{\sqrt{5}} }$$
4. $$\displaystyle \ln { \frac{9}{5} }$$
5. $$\displaystyle \frac{1}{2} \ln {12}$$
6. Solution C
7. $$\displaystyle \int_4^6 \frac{x - 1}{x^2- 4} \, dx =$$
1. $$\displaystyle \frac{7 \ln {2} + 3 \ln {3}}{4}$$
2. $$\displaystyle \frac{7 \ln {2} - 3 \ln {3}}{4}$$
3. $$\displaystyle \frac{7}{4} \ln {\frac{3}{2} }$$
4. $$\displaystyle \frac{4}{7} \ln { \frac{3}{2} }$$
5. $$\displaystyle \frac{7 \ln {2} - 2 \ln {3}}{4}$$
8. Solution B
9. $$\displaystyle \int x \cos{x^2} \, dx =$$
1. $$\displaystyle x \sin{x^2} + C$$
2. $$\displaystyle \frac{1}{2} \sin{x^2} + C$$
3. $$\displaystyle -\frac{1}{2} \sin{x^2} + C$$
4. $$\displaystyle \cos{x^2} + C$$
5. $$\displaystyle \sin{x^2} + C$$
10. Solution B
11. $$\displaystyle \int_1^{e^2} x^2 \ln {x^2} \, dx =$$
1. $$\displaystyle \frac{e^6 - 2}{9}$$
2. $$\displaystyle \frac{e^6 + 2}{9}$$
3. $$\displaystyle \frac{10e^6 - 2}{9}$$
4. $$\displaystyle \frac{10e^6 + 2}{9}$$
5. $$\displaystyle \frac{10e^6 + 1}{9}$$
12. Solution D
13. $$\displaystyle \int \frac{x^2 + 5x}{x^2 + 5x + 6} \, dx =$$
1. $$\displaystyle -6 \ln {\left| \frac{x + 2}{x + 3} \right| } + C$$
2. $$\displaystyle x - 6 \ln { \left| \frac{ x + 2}{x + 3} \right| } + C$$
3. $$\displaystyle -5 \ln { \left| \frac{ x + 2}{x + 3} \right| } + C$$
4. $$\displaystyle - \ln { \left| \frac{ x + 2}{x + 3} \right| } + C$$
5. $$\displaystyle \frac{x}{6} + C$$
14. Solution B
15. $$\displaystyle \int \frac{(x + 2)^3}{x^2 + 2x} \, dx =$$
1. $$\displaystyle 4 \ln {|x|} + C$$
2. $$\displaystyle \frac{1}{2} x^2 + 4x + \ln {|x|} + C$$
3. $$\displaystyle \frac{1}{2} x^2 + 2x + 4 \ln {|x|} + C$$
4. $$\displaystyle \frac{1}{2} x^2 + 4x + 4 \ln {|x|} + C$$
5. $$\displaystyle \frac{1}{2} x^2 + 4 \ln {|x|} + C$$
16. Solution D
17. $$\displaystyle \int_1^{\infty} \frac{2}{\sqrt[3]{x}} \, dx =$$
1. $$\displaystyle -\infty$$
2. $$\displaystyle - \frac{2}{3}$$
3. 0
4. $$\displaystyle \frac{2}{3}$$
5. $$\displaystyle \infty$$
18. Solution E
19. $$\displaystyle \int_0^1 \frac{dx}{\sqrt[3]{x - 1}} =$$
1. $$\displaystyle -\infty$$
2. $$\displaystyle -\frac{3}{2}$$
3. $$\displaystyle 0$$
4. $$\displaystyle \frac{3}{2}$$
5. $$\displaystyle \infty$$
20. Solution B
21. $$\displaystyle \int_1^r \frac{dx}{x \ln{x}}$$ where $$r$$ is a real number greater than 1 =
1. 0
2. 1
3. $$\displaystyle \ln {r}$$
4. $$\displaystyle -\infty$$
5. $$\displaystyle \infty$$
22. Solution E
23. When $$n$$ is an integer greater than 1, $$\displaystyle \int_0^1 \frac{dx}{x^n} =$$
1. 0
2. $$\displaystyle \frac{1}{-n + 1}$$
3. 1
4. $$\displaystyle -\infty$$
5. $$\displaystyle \infty$$
24. Solution E
25. When $$n$$ is an integer greater than 1, $$\displaystyle \int_1^{\infty} \frac{dx}{x^n} =$$
1. 0
2. 1
3. $$\displaystyle \frac{1}{n - 1}$$
4. $$\displaystyle -\infty$$
5. $$\displaystyle \infty$$
26. Solution C
27. $$\displaystyle \int_1^3 x^3 \ln {x } \, dx =$$
1. $$\displaystyle 27 \ln { 3} - 12$$
2. $$\displaystyle 27 \ln { 3} - \frac{27}{2}$$
3. $$\displaystyle \frac{81}{4} \ln { 3} - \frac{81}{16}$$
4. $$\displaystyle \frac{81}{4} \ln { 3} - 5$$
5. $$\displaystyle 27 \ln {3}$$
28. Solution D
29. $$\displaystyle \int_0^{\frac{\pi}{4}} x \sec^2 {x} \, dx =$$
1. $$\displaystyle \frac{\pi}{4}$$
2. $$\displaystyle \frac{\pi}{4} + 1$$
3. $$\displaystyle \frac{\pi}{4} - \frac{3}{2} \ln {2}$$
4. $$\displaystyle \frac{\pi}{4} - \frac{1}{2} \ln {2}$$
30. Solution D
31. $$\displaystyle \int_1^2 \frac{x^3 dx}{\sqrt{x^2 - 1}} =$$
1. 0
2. $$\displaystyle \sqrt{3}$$
3. $$\displaystyle 2\sqrt{3}$$
4. $$\displaystyle 4\sqrt{3}$$
5. $$\displaystyle \infty$$
32. Solution C
33. $$\displaystyle \int_1^2 \frac{x^4 + 1}{x^3 + x^2} \, dx =$$
1. $$\displaystyle \ln {\frac{9}{8} + 1}$$
2. $$\displaystyle \ln {\frac{8}{9} + 1}$$
3. $$\displaystyle \ln {\frac{9}{8}} + \frac{1}{2}$$
4. $$\displaystyle \ln {\frac{8}{9}} + \frac{1}{2}$$
5. $$\displaystyle \ln {\frac{9}{8}}$$
34. Solution A
35. Find the area under the curve $$\displaystyle f(x) = \frac{1}{ x^2 - x}$$ for $$x \geq 2$$.
1. $$\displaystyle - \ln {2}$$
2. $$\displaystyle \ln {2}$$
3. 1.25
4. $$\displaystyle e^2$$
5. $$\displaystyle \infty$$
36. Solution B
37. Find $$\displaystyle \lim_{n \to \infty} \int_0^2 \frac{dx}{\sqrt[n]{x}}$$ for $$n > 1$$.
1. 0
2. 1
3. 2
4. $$\displaystyle 2\sqrt{2}$$
5. $$\displaystyle \infty$$
38. Solution C
39. The region bounded by the graph of $$\displaystyle y = \tan {x}$$, the x-axis, and the line $$\displaystyle x = \frac{\pi}{4}$$ is the base of a solid. Find the volume of the solid if cross-sections perpendicular to the x-axis are squares.
1. 1
2. $$\displaystyle 1 - \frac{\pi}{4}$$
3. $$\displaystyle 4 - \pi$$
4. $$\displaystyle 4 - \frac{\pi}{4}$$
5. $$\displaystyle \sqrt{3} - \frac{\pi }{3}$$
40. Solution B
41. If $$\displaystyle f(x) = \frac{1}{\sqrt{x^2 - 1}}$$, then $$\displaystyle \int_1^2 xf(x) \, dx =$$
1. $$\displaystyle \sqrt{3}$$
2. $$\displaystyle \sqrt{3} - 1$$
3. $$\displaystyle 1/\sqrt{3}$$
4. 1
5. $$\displaystyle \infty$$
42. Solution A
43. $$\displaystyle \int x^3 f''\left( x^2 \right) \, dx =$$
1. $$\displaystyle \frac{1}{2} \left( x^2f(x^2) - f(x^2) \right) + C$$
2. $$\displaystyle \frac{1}{2} \left( x^2f'(x^2) - f(x^2) \right) + C$$
3. $$\displaystyle x^2f'(x^2) - f(x^2) + C$$
4. $$\displaystyle 2 \left( x^2f'(x^2) - f(x^2) \right) + C$$
5. $$\displaystyle \frac{1}{2} \left( x^2f''(x^2) - f'(x^2) \right) + C$$
44. Solution B
45. Find the area between the x-axis and the curve $$\displaystyle f(x) = \frac{1}{ x^2 + x - 6}$$ on the interval [-2, 0].
1. $$\displaystyle \ln { \frac{3}{2}}$$
2. $$\displaystyle \frac{1}{5} \ln { \frac{3}{2}}$$
3. $$\displaystyle \frac{1}{5} \ln {6}$$
4. $$\displaystyle \ln {6}$$
5. undefined
46. Solution C
47. $$\displaystyle f(x) = 2x^2, g(x) = \csc{x}$$. Find $$\displaystyle \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} f \left( g(x) \right) \, dx.$$
1. -2
2. -1
3. 1
4. $$\displaystyle \sqrt{3}$$
5. 2
48. Solution E
49. $$\displaystyle \int \frac{dx}{5 + 5 \cos{x}} =$$
1. $$\displaystyle 5x + 5 \sin{x} + C$$
2. $$\displaystyle \frac{1}{5x + 5 \sin{x} } + C$$
3. $$\displaystyle \frac{1}{5} \tan {\frac{x}{2}} + C$$
4. $$\displaystyle \ln {(5 + 5\cos{x})} + C$$
5. $$\displaystyle \frac{1}{5} \left( x + \ln {\left| \sec{x} + \tan{x} \right|} \right) +C$$
50. Solution C
51. $$\displaystyle \int_1^{\infty} \frac{ dx}{x \left( \ln {x} + 1 \right)^2} =$$
1. $$\displaystyle \frac{1}{2}$$
2. 1
3. 2
4. 3
5. $$\displaystyle \infty$$
52. Solution B
53. $$\displaystyle \int_0^2 \sqrt[5]{1 - x} \, dx$$ =
1. 0
2. $$\displaystyle \frac{5}{6}$$
3. $$\displaystyle \frac{5}{4}$$
4. $$\displaystyle \frac{5}{2}$$
5. undefined
54. Solution A
55. Find the y-intercept of the line tangent to the graph of $$\displaystyle xy^2 + \ln { x} = y + 6$$ at the point (1, 3).
1. $$\displaystyle -\frac{23}{5}$$
2. -2
3. 1
4. $$\displaystyle \frac{7}{5}$$
5. 5
56. Solution E