BC Calculus : MC Integration Practice

  1. \( \displaystyle \int_{-1}^{1} (x^2 - x - 1) \, dx = \)
    1. \(\frac{2}{3}\)
    2. 0
    3. \(-\frac{4}{3}\)
    4. -2
    5. -1
    2. Solution C
  2. \( \displaystyle \int_{1}^{2} \frac{3x - 1}{3x} \, dx = \)
    1. \(\frac{3}{4}\)
    2. \( 1 - \frac{1}{3}\ln{2} \)
    3. \(1 - \ln{ 2}\)
    4. -\(\frac{1}{3}\ln{ 2}\)
    5. 1
    3. Solution B
  3. \( \displaystyle \int_{0}^{3} \frac{dt}{\sqrt{4 - t}} = \)
    1. 1
    2. -2
    3. 4
    4. -1
    5. 2
    4. Solution E
  4. \( \displaystyle \int_{-1}^{0} \sqrt{3u + 4} \, du = \)
    1. 2
    2. \(\frac{14}{9}\)
    3. \(\frac{4}{3}\)
    4. 6
    5. \(\frac{7}{2}\)
    5. Solution B
  5. \( \displaystyle \int_{2}^{3} \frac{dy}{2y - 3} = \)
    1. \( \ln{ 3}\)
    2. \(\frac{1}{2} \ln{ \left( \frac{3}{2} \right) } \)
    3. \(\frac{16}{9}\)
    4. \(\frac{1}{2}\ln{3}\)
    5. \(\sqrt{3} - 1\)
    6. Solution D
  6. \( \displaystyle \int_{0}^{\sqrt{3}} \frac{x dx}{ \sqrt{4 - x^2}} = \)
    1. 1
    2. \( \frac{\pi }{6}\)
    3. \( \frac{ \pi}{3} \)
    4. -1
    5. 2
    7. Solution A
  7. \( \displaystyle \int_{0}^{1} (2t - 1)^3 \, dt = \)
    1. \frac{1}{4}
    2. 6
    3. \(\frac{1}{2}\)
    4. 0
    5. 4
    8. Solution D
  8. \( \displaystyle \int_{0}^{1} \frac{dx}{\sqrt{4 - x^2}} = \)
    1. \(\frac{ \pi}{3}\)
    2. \(2 -\sqrt{3}\)
    3. \( \frac{ \pi}{12}\)
    4. \(2( \sqrt{3} - 2)\)
    5. \( \frac{\pi }{6}\)
    9. Solution E
  9. \( \displaystyle \int_{4}^{9} \frac{(2 + x) dx}{2 \sqrt{x}} = \)
    1. \(\frac{25}{3}\)
    2. \(\frac{41}{3}\)
    3. \(\frac{100}{3}\)
    4. \(\frac{5}{3}\)
    5. \(\frac{1}{3}\)
    10. Solution A
  10. \( \displaystyle \int_{-3}^{3} \frac{dx}{9 + x^2} = \)
    1. \(\frac{ \pi}{2}\)
    2. 0
    3. \( \frac{\pi }{6}\)
    4. \(- \frac{ \pi}{2} \)
    5. \(\frac{ \pi}{3}\)
    11. Solution C
  11. \( \displaystyle \int_{0}^{1}e^{-x} \, dx = \)
    1. \( \frac{1}{e} - 1\)
    2. \(1 - e\)
    3. \(- \frac{1}{e}\)
    4. \(1 - \frac{1}{e} \)
    5. \( \frac{1}{e}\)
    12. Solution D
  12. \( \displaystyle \int_{0}^{1} xe^{x^2} \, dx = \)
    1. \(e - 1\)
    2. \(\frac{1}{2}(e - 1)\)
    3. \(2(e - 1)\)
    4. \(\frac{e}{2}\)
    5. \(\frac{e}{2} - 1\)
    13. Solution B
  13. \( \displaystyle \int_{0}^{ \frac{ \pi}{4} }\sin{(2\theta )} \, d\theta = \)
    1. 2
    2. \(\frac{1}{2}\)
    3. -1
    4. \(-\frac{1}{2}\)
    5. -2
    14. Solution B
  14. \( \displaystyle \int_{1}^{2} \frac{dz}{3 - z} = \)
    1. \(-\ln{2} \)
    2. \(\frac{3}{4} \)
    3. \(2(\sqrt{2} - 1) \)
    4. \(\frac{1}{2}\ln{ 2} \)
    5. \(\ln{ 2} \)
    15. Solution E
  15. \( \displaystyle \int_{1}^{e} \ln{ y} \, dy = \)
    1. \(2e + 1\)
    2. \(\frac{1}{2}\)
    3. 1
    4. \(e - 1\)
    5. -1
    16. Solution C
  16. \( \displaystyle \int_{-4}^{4} \sqrt{16 - x^2} \, dx = \)
    1. \(8 \pi \)
    2. \(4 \pi \)
    3. 4
    4. 8
    5. none of these
    17. Solution A
  17. \( \displaystyle \int_{0}^{ \pi } \cos^2{\theta }\sin{ \theta } \, d\theta = \)
    1. \(-\frac{2}{3}\)
    2. \(\frac{1}{3}\)
    3. 1
    4. \(\frac{2}{3}\)
    5. 0
    18. Solution D
  18. \( \displaystyle \int_{1}^{e} \frac{\ln{ x}}{ x} \, dx = \)
    1. \(\frac{1}{2}\)
    2. \(\frac{1}{2}(e^2 - 1)\)
    3. 0
    4. 1
    5. \(e - 1\)
    19. Solution A
  19. \( \displaystyle \int_{0}^{1} xe^x \, dx = \)
    1. -1
    2. \(e + 1 \)
    3. 1
    4. \(e - 1 \)
    5. \(\frac{1}{2}(e - 1) \)
    20. Solution C
  20. \( \displaystyle \int_{0}^{ \frac{\pi }{6}} \frac{\cos {\theta } }{1 + 2 \sin {\theta}} \, d\theta = \)
    1. \(\ln{ 2}\)
    2. \(\frac{3}{8}\)
    3. \(-\frac{1}{2}\ln{ 2}\)
    4. \(\frac{3}{2}\)
    5. \(\ln{ \sqrt{2}}\)
    21. Solution E
  21. \( \displaystyle \int_{\sqrt{2}}^{2} \frac{u}{u^2 - 1} \, du = \)
    1. \(\ln{ \sqrt{3}}\)
    2. \(\frac{8}{9}\)
    3. \(\ln{ \frac{3}{2}}\)
    4. \(\ln{ 3}\)
    5. \(1 - \sqrt{3}\)
    22. Solution A
  22. \( \displaystyle \int_{\sqrt{2}}^{2} \frac{u}{(u^2 - 1)^2} \, du = \)
    1. \(-\frac{1}{3}\)
    2. \(-\frac{2}{3}\)
    3. \(\frac{2}{3}\)
    4. -1
    5. \(\frac{1}{3}\)
    23. Solution E
  23. \( \displaystyle \int_{ \frac{ \pi}{12}}^{ \frac{ \pi}{4}} \frac{\cos {(2x)} }{\sin^2 {(2x)}} = \)
    1. \(-\frac{1}{4}\)
    2. 1
    3. \(\frac{1}{2}\)
    4. \(-\frac{1}{2}\)
    5. -1
    24. Solution C
  24. \( \displaystyle \int_{0}^{1} \frac{e^{-x} + 1}{ e^{-x}} \, dx = \)
    1. \(e\)
    2. \(2 + e\)
    3. \( \frac{1}{e}\)
    4. \(1 + e\)
    5. \(e - 1\)
    25. Solution A
  25. \( \displaystyle \int_{0}^{1} \frac{ e^x dx}{e^x + 1} = \)
    1. \(\ln{ 2}\)
    2. \(e\)
    3. \(1 + e\)
    4. \(-\ln{ 2|}\)
    5. \(\ln{ \left( \frac{e + 1}{2} \right) }\)
    26. Solution E
  26. If the substitution \( u = \sqrt{x + 1}\) is used, then \( \displaystyle \int_{0}^{3} \frac{dx}{x\sqrt{x + 1}}\) is equivalent to
    1. \( \displaystyle \int_{1}^{2} \frac{du}{u^2 - 1}\)
    2. \( \displaystyle \int_{1}^{2} \frac{2du}{u^2 - 1}\)
    3. \(2 \displaystyle \int_{0}^{3} \frac{du}{(u - 1)(u + 1)}\)
    4. \(2 \displaystyle \int_{1}^{2} \frac{du}{u(u^2 - 1)}\)
    5. \(2 \displaystyle \int_{0}^{3} \frac{du}{u(u - 1)}\)
    27. Solution B