Advanced Integration Techniques: Get Stronger Answer Key

Integration by Parts

  1. [latex]u={x}^{3}[/latex]
  2. [latex]u={y}^{3}[/latex]
  3. [latex]u=\sin\left(2x\right)[/latex]
  4. [latex]\text{-}x+x\text{ln}x+C[/latex]
  5. [latex]x{\tan}^{-1}x-\frac{1}{2}\text{ln}\left(1+{x}^{2}\right)+C[/latex]
  6. [latex]-\frac{1}{2}x\cos\left(2x\right)+\frac{1}{4}\sin\left(2x\right)+C[/latex]
  7. [latex]{e}^{\text{-}x}\left(-1-x\right)+C[/latex]
  8. [latex]2x\cos{x}+\left(-2+{x}^{2}\right)\sin{x}+C[/latex]
  9. [latex]\frac{1}{2}\left(1+2x\right)\left(-1+\text{ln}\left(1+2x\right)\right)+C[/latex]
  10. [latex]\frac{1}{2}{e}^{x}\left(\text{-}\cos{x}+\sin{x}\right)+C[/latex]
  11. [latex]-\frac{{e}^{\text{-}{x}^{2}}}{2}+C[/latex]
  12. [latex]-\frac{1}{2}x\cos\left[\text{ln}\left(2x\right)\right]+\frac{1}{2}x\sin\left[\text{ln}\left(2x\right)\right]+C[/latex]
  13. [latex]2x - 2x\text{ln}x+x{\left(\text{ln}x\right)}^{2}+C[/latex]
  14. [latex]\left(\text{-}\frac{{x}^{3}}{9}+\frac{1}{3}{x}^{3}\text{ln}x\right)+C[/latex]
  15. [latex]-\frac{1}{2}\sqrt{1 - 4{x}^{2}}+x{\cos}^{-1}\left(2x\right)+C[/latex]
  16. [latex]\text{-}\left(-2+{x}^{2}\right)\cos{x}+2x\sin{x}+C[/latex]
  17. [latex]\text{-}x\left(-6+{x}^{2}\right)\cos{x}+3\left(-2+{x}^{2}\right)\sin{x}+C[/latex]
  18. [latex]\frac{1}{2}x\left(\text{-}\sqrt{1-\frac{1}{{x}^{2}}}+x\cdot {\sec}^{-1}x\right)+C[/latex]
  19. [latex]\text{-}\text{cosh}x+x\text{sinh}x+C[/latex]
  20. [latex]\frac{1}{4}-\frac{3}{4{\text{e}}^{2}}[/latex]
  21. [latex]2[/latex]
  22. [latex]2\pi[/latex]
  23. [latex]-2+\pi[/latex]
  24. [latex]\text{-}\sin\left(x\right)+\text{ln}\left[\sin\left(x\right)\right]\sin{x}+C[/latex]
  25. Answers vary
    1. [latex]\frac{2}{5}\left(1+x\right){\left(-3+2x\right)}^{\frac{3}{2}}+C[/latex]
    2. [latex]\frac{2}{5}\left(1+x\right){\left(-3+2x\right)}^{\frac{3}{2}}+C[/latex]
  26. Do not use integration by parts. Choose [latex]u[/latex] to be [latex]\text{ln}x[/latex], and the integral is of the form [latex]\displaystyle\int {u}^{2}du[/latex].
  27. Do not use integration by parts. Let [latex]u={x}^{2}-3[/latex], and the integral can be put into the form [latex]\displaystyle\int {e}^{u}du[/latex].
  28. Do not use integration by parts. Choose [latex]u[/latex] to be [latex]u=3{x}^{3}+2[/latex] and the integral can be put into the form [latex]\displaystyle\int \sin\left(u\right)du[/latex].

  29. The area under graph is [latex]0.39535[/latex].


    This figure is the graph of y=e^-x sin(pi*x). The curve begins in the third quadrant at x=0.5, increases through the origin, reaches a high point between 0.5 and 0.75, then decreases, passing through x=1.

  30. [latex]2\pi e[/latex]
  31. [latex]2.05[/latex]
  32. [latex]12\pi[/latex]
  33. [latex]8{\pi }^{2}[/latex]

Trigonometric Integrals

    1. [latex]\dfrac{{\sin}^{4}x}{4}+C[/latex]
    2. [latex]\dfrac{1}{12}{\tan}^{6}\left(2x\right)+C[/latex]
    3. [latex]{\sec}^{2}\left(\dfrac{x}{2}\right)+C[/latex]
    4. [latex]\text{-}\cos{x}+\dfrac{1}{3}\cos^{2}x+C[/latex]
    5. [latex]-\dfrac{1}{2}{\cos}^{2}x+C[/latex] or [latex]\dfrac{1}{2}{\sin}^{2}x+C[/latex]
    6. [latex]-\dfrac{1}{3}\cos^{3}x+\dfrac{2}{5}\cos^{5}x-\dfrac{1}{7}\cos^{7}x+C[/latex]
    7. [latex]\dfrac{2}{3}{\left(\sin{x}\right)}^{\dfrac{3}{2}}+C[/latex]
    8. [latex]\sec{x}+C[/latex]
    9. [latex]\dfrac{1}{2}\sec{x}\tan{x}-\dfrac{1}{2}\text{ln}\left(\sec{x}+\tan{x}\right)+C[/latex]
    10. [latex]\dfrac{2\tan{x}}{3}+\dfrac{1}{3}\sec{\left(x\right)}^{2}\tan{x}[/latex] [latex]=\tan{x}+\dfrac{{\tan}^{3}x}{3}+C[/latex]
    11. [latex]\text{-}\text{ln}|\cot{x}+\csc{x}|+C[/latex]
    12. [latex]\dfrac{{\sin}^{3}\left(ax\right)}{3a}+C[/latex]
    13. [latex]\dfrac{\pi }{2}[/latex]
    14. [latex]\dfrac{x}{2}+\dfrac{1}{12}\sin\left(6x\right)+C[/latex]
    15. [latex]x+C[/latex]
    16. [latex]0[/latex]
    17. [latex]0[/latex]
    18. [latex]0[/latex]
    19. [latex]\text{Approximately 0.239}[/latex]
    20. [latex]\sqrt{2}[/latex]
    21. [latex]1.0[/latex]
    22. [latex]0[/latex]
    23. [latex]\dfrac{3\theta }{8}-\dfrac{1}{4\pi }\sin\left(2\pi \theta \right)+\dfrac{1}{32\pi }\sin\left(4\pi \theta \right)+C=f\left(x\right)[/latex]
    24. [latex]\text{ln}\left(\sqrt{3}\right)[/latex]
    25. [latex]{\displaystyle\int }_{\text{-}\pi }^{\pi }\sin\left(2x\right)\cos\left(3x\right)dx=0[/latex]
    26. [latex]\sqrt{\tan\left(x\right)}x\left(\dfrac{8\tan{x}}{21}+\dfrac{2}{7}\sec{x}^{2}\tan{x}\right)+C=f\left(x\right)[/latex]
    27. The second integral is more difficult because the first integral is simply a u-substitution type.
    28. [latex]9{\tan}^{2}\theta[/latex]
    29. [latex]{a}^{2}{\text{cosh}}^{2}\theta[/latex]
    30. [latex]4{\left(x-\dfrac{1}{2}\right)}^{2}[/latex]
    31. [latex]\text{-}{\left(x+1\right)}^{2}+5[/latex]
    32. [latex]\text{ln}|x+\sqrt{\text{-}{a}^{2}+{x}^{2}}|+C[/latex]
    33. [latex]\dfrac{1}{3}\text{ln}|\sqrt{9{x}^{2}+1}+3x|+C[/latex]
    34. [latex]-\dfrac{\sqrt{1-{x}^{2}}}{x}+C[/latex]
    35. [latex]9\left[\dfrac{x\sqrt{{x}^{2}+9}}{18}+\dfrac{1}{2}ln|\dfrac{\sqrt{{x}^{2}+9}}{3}+\dfrac{x}{3}|\right]+C[/latex]
    36. [latex]-\dfrac{1}{3}\sqrt{9-{\theta }^{2}}\left(18+{\theta }^{2}\right)+C[/latex]
    37. [latex]-\dfrac{\sqrt{1+{x}^{2}}}{x}+C[/latex]
    38. [latex]\dfrac{1}{8}\left(x\left(5 - 2{x}^{2}\right)\sqrt{1-{x}^{2}}+3\text{arcsin}x\right)+C[/latex]
    39. [latex]\text{ln}x-\text{ln}|1+\sqrt{1-{x}^{2}}|+C[/latex]
    40. [latex]-\dfrac{\sqrt{-1+{x}^{2}}}{x}+\text{ln}|x+\sqrt{-1+{x}^{2}}|+C[/latex]
    41. [latex]-\dfrac{\sqrt{1+{x}^{2}}}{x}+\text{arcsinh}x+C[/latex]
    42. [latex]-\dfrac{1}{1+x}+C[/latex]
    43. [latex]\dfrac{2\sqrt{-10+x}\sqrt{x}\text{ln}|\sqrt{-10+x}+\sqrt{x}|}{\sqrt{\left(10-x\right)x}}+C[/latex]
    44. [latex]\dfrac{9\pi }{2}[/latex]; area of a semicircle with radius 3
    45. [latex]\text{arcsin}\left(x\right)+C[/latex] is the common answer.
    46. [latex]\dfrac{1}{2}\text{ln}\left(1+{x}^{2}\right)+C[/latex] is the result using either method.
    47. Use trigonometric substitution. Let [latex]x=\sec\left(\theta \right)[/latex].
    48. [latex]4.367[/latex]
    49. [latex]\dfrac{{\pi }^{2}}{8}+\dfrac{\pi }{4}[/latex]
    50. [latex]y=\dfrac{1}{16}\text{ln}|\dfrac{x+8}{x - 8}|+3[/latex]
    51. [latex]24.6[/latex] m³
    52. [latex]\dfrac{2\pi }{3}[/latex]

Partial Fractions

  1. [latex]-\dfrac{2}{x+1}+\dfrac{5}{2\left(x+2\right)}+\dfrac{1}{2x}[/latex]
  2. [latex]\dfrac{1}{{x}^{2}}+\dfrac{3}{x}[/latex]
  3. [latex]2{x}^{2}+4x+8+\dfrac{16}{x - 2}[/latex]
  4. [latex]-\dfrac{1}{{x}^{2}}-\dfrac{1}{x}+\dfrac{1}{x - 1}[/latex]
  5. [latex]-\dfrac{1}{2\left(x - 2\right)}+\dfrac{1}{2\left(x - 1\right)}-\dfrac{1}{6x}+\dfrac{1}{6\left(x - 3\right)}[/latex]
  6. [latex]\dfrac{1}{x - 1}+\dfrac{2x+1}{{x}^{2}+x+1}[/latex]
  7. [latex]\dfrac{2}{x+1}+\dfrac{x}{{x}^{2}+4}-\dfrac{1}{{\left({x}^{2}+4\right)}^{2}}[/latex]
  8. [latex]\text{-}\text{ln}|2-x|+2\text{ln}|4+x|+C[/latex]
  9. [latex]\dfrac{1}{2}\text{ln}|4-{x}^{2}|+C[/latex]
  10. [latex]2\left(x+\dfrac{1}{3}\text{arctan}\left(\dfrac{1+x}{3}\right)\right)+C[/latex]
  11. [latex]2\text{ln}|x|-3\text{ln}|1+x|+C[/latex]
  12. [latex]\dfrac{1}{16}\left(\text{-}\dfrac{4}{-2+x}-\text{ln}|-2+x|+\text{ln}|2+x|\right)+C[/latex]
  13. [latex]\dfrac{1}{30}\left(-2\sqrt{5}\text{arctan}\left[\dfrac{1+x}{\sqrt{5}}\right]+2\text{ln}|-4+x|-\text{ln}|6+2x+{x}^{2}|\right)+C[/latex]
  14. [latex]-\dfrac{3}{x}+4\text{ln}|x+2|+x+C[/latex]
  15. [latex]\text{-}\text{ln}|3-x|+\dfrac{1}{2}\text{ln}|{x}^{2}+4|+C[/latex]
  16. [latex]\text{ln}|x - 2|-\dfrac{1}{2}\text{ln}|{x}^{2}+2x+2|+C[/latex]
  17. [latex]\text{-}x+\text{ln}|1-{e}^{x}|+C[/latex]
  18. [latex]\dfrac{1}{5}\text{ln}|\dfrac{\cos{x}+3}{\cos{x} - 2}|+C[/latex]
  19. [latex]\dfrac{1}{2 - 2{e}^{2t}}+C[/latex]
  20. [latex]2\sqrt{1+x}-2\text{ln}|1+\sqrt{1+x}|+C[/latex]
  21. [latex]\text{ln}|\dfrac{\sin{x}}{1-\sin{x}}|+C[/latex]
  22. [latex]\dfrac{\sqrt{3}}{4}[/latex]
  23. [latex]x-\text{ln}\left(1+{e}^{x}\right)+C[/latex]
  24. [latex]6{x}^{\dfrac{1}{6}}-3{x}^{\dfrac{1}{3}}+2\sqrt{x}-6\text{ln}\left(1+{x}^{\dfrac{1}{6}}\right)+C[/latex]
  25. [latex]\dfrac{4}{3}\pi \text{arctanh}\left[\dfrac{1}{3}\right]=\dfrac{1}{3}\pi \text{ln}4[/latex]
  26. [latex]x=\text{-}\text{ln}|t - 3|+\text{ln}|t - 4|+\text{ln}2[/latex]
  27. [latex]x=\text{ln}|t - 1|-\sqrt{2}\text{arctan}\left(\sqrt{2}t\right)-\dfrac{1}{2}\text{ln}\left({t}^{2}+\dfrac{1}{2}\right)+\sqrt{2}\text{arctan}\left(2\sqrt{2}\right)+\dfrac{1}{2}\text{ln}4.5[/latex]
  28. [latex]\dfrac{2}{5}\pi \text{ln}\dfrac{28}{13}[/latex]
  29. [latex]\dfrac{\text{arctan}\left[\dfrac{-1+2x}{\sqrt{3}}\right]}{\sqrt{3}}+\dfrac{1}{3}\text{ln}|1+x|-\dfrac{1}{6}\text{ln}|1-x+{x}^{2}|+C[/latex]
  30. [latex]2.0[/latex] in.²
  31. [latex]3{\left(-8+x\right)}^{\dfrac{1}{3}}[/latex] [latex]-2\sqrt{3}\text{arctan}\left[\dfrac{-1+{\left(-8+x\right)}^{\dfrac{1}{3}}}{\sqrt{3}}\right][/latex] [latex]-2\text{ln}\left[2+{\left(-8+x\right)}^{\dfrac{1}{3}}\right][/latex] [latex]+\text{ln}\left[4 - 2{\left(-8+x\right)}^{\dfrac{1}{3}}+{\left(-8+x\right)}^{\dfrac{2}{3}}\right]+C[/latex]

Other Strategies for Integration

  1. [latex]\dfrac{1}{2}\text{ln}|{x}^{2}+2x+2|+2\text{arctan}\left(x+1\right)+C[/latex]
  2. [latex]{\text{cosh}}^{-1}\left(\dfrac{x+3}{3}\right)+C[/latex]
  3. [latex]\dfrac{{2}^{{x}^{2}-1}}{\text{ln}2}+C[/latex]
  4. [latex]\text{arcsin}\left(\dfrac{y}{2}\right)+C[/latex]
  5. [latex]-\dfrac{1}{2}\csc\left(2w\right)+C[/latex]
  6. [latex]9 - 6\sqrt{2}[/latex]
  7. [latex]2-\dfrac{\pi }{2}[/latex]
  8. [latex]\dfrac{1}{12}{\tan}^{4}\left(3x\right)-\dfrac{1}{6}{\tan}^{2}\left(3x\right)+\dfrac{1}{3}\text{ln}|\sec\left(3x\right)|+C[/latex]
  9. [latex]2\cot\left(\dfrac{w}{2}\right)-2\csc\left(\dfrac{w}{2}\right)+w+C[/latex]
  10. [latex]\dfrac{1}{5}\text{ln}|\dfrac{2\left(5+4\sin{t} - 3\cos{t}\right)}{4\cos{t}+3\sin{t}}|[/latex]
  11. [latex]6{x}^{\dfrac{1}{6}}-3{x}^{\dfrac{1}{3}}+2\sqrt{x}-6\text{ln}\left[1+{x}^{\dfrac{1}{6}}\right]+C[/latex]
  12. [latex]\text{-}{x}^{3}\cos{x}+3{x}^{2}\sin{x}+6x\cos{x} - 6\sin{x}+C[/latex]
  13. [latex]\dfrac{1}{2}\left({x}^{2}+\text{ln}|1+{e}^{\text{-}{x}^{2}}|\right)+C[/latex]
  14. [latex]2\text{arctan}\left(\sqrt{x - 1}\right)+C[/latex]
  15. [latex]0.5=\dfrac{1}{2}[/latex]
  16. [latex]8.0[/latex]
  17. [latex]\dfrac{1}{3}\text{arctan}\left(\dfrac{1}{3}\left(x+2\right)\right)+C[/latex]
  18. [latex]\dfrac{1}{3}\text{arctan}\left(\dfrac{x+1}{3}\right)+C[/latex]
  19. [latex]\text{ln}\left({e}^{x}+\sqrt{4+{e}^{2x}}\right)+C[/latex]
  20. [latex]\text{ln}x-\dfrac{1}{6}\text{ln}\left({x}^{6}+1\right)-\dfrac{\text{arctan}\left({x}^{3}\right)}{3{x}^{3}}+C[/latex]
  21. [latex]\text{ln}|x+\sqrt{16+{x}^{2}}|+C[/latex]
  22. [latex]-\dfrac{1}{4}\cot\left(2x\right)+C[/latex]
  23. [latex]\dfrac{1}{2}\text{arctan}10[/latex]
  24. [latex]1276.14[/latex]
  25. [latex]7.21[/latex]
  26. [latex]\sqrt{5}-\sqrt{2}+\text{ln}|\dfrac{2+2\sqrt{2}}{1+\sqrt{5}}|[/latex]
  27. [latex]\dfrac{1}{3}\text{arctan}\left(3\right)\approx 0.416[/latex]