MIT Integration Bee

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Brain training.

Qualifier Test

2010

  1. \[ \int_0^{\pi/2} \sin (x) \sin (2x) \sin (3x) dx \]

    Solution

    Details and summary

    Note with summary: note primary This is a summary

  2. \[ \int_0^{\pi/2} \sin^3(2x) \cos(x) dx \]

  3. \[ \int (x+1)^2 (x-1)^{1/3} dx \]

  4. \[ \int x \log \left(1 + \frac{1}{x}\right) dx \]

  5. \[ \int_0^1 \sin^2(\log x) dx \]

  6. \[ \int \frac{1}{1 + 3 e^x} dx \]

  7. \[ \int_{\pi/4}^{\pi/3} \frac{dx}{\sin^3(x) \cos^5(x)}dx \]

  8. \[ \int_1^\infty \frac{dx}{x \sqrt{x^4 - 1}}dx \]

  9. \[ \int \frac{dx}{x(x^5 + 1)}dx \]

  10. \[ \int_0^{\pi/4} \sqrt{\tan(x)} dx \]

  11. \[ \int_0^1 \frac{\log(1 + x)}{1 + x^2} dx \]

  12. \[ \int_{64}^{729} \frac{x^{1/2}}{x^{1/2} - x^{1/3}} dx \]

  13. \[ \int x^x (1 + \log x) dx \]

  14. \[ \int_0^1 x^{13/2} \sqrt{1 + x^{5/2}} dx \]

  15. \[ \int_1^\infty \frac{dx}{(x^2 + 1)^2}dx \]

  16. \[ \int_0^1 \frac{dx}{x^4 - 13 x^2 + 36}dx \]

  17. \[ \int \frac{\log(\log x)}{x} dx \]

  18. \[ \int \frac{1 + \cot x}{1 - \cot x} dx \]

  19. \[ \int \frac{\cos(x) + x \sin(x)}{x (x + \cos x)} dx \]

  20. \[ \int_0^{\pi/2} \frac{dx}{\sin(x) + \sec(x)}dx \]

  21. \[ \int_0^\infty \frac{dx}{\sqrt{1 + e^x + e^{2x}}}dx \]

  22. \[ \int_0^1 x^3 e^{x^2} dx \]

  23. \[ \int_0^1 \sqrt{1 + x \sqrt{1 + x \sqrt{1 + x \sqrt{\cdots}}}} dx \]

  24. \[ \int \left( \frac{1}{\log x} - \frac{1}{(\log x)^2} \right) dx \]

  25. \[ \int_1^2 (x-1)^{1/2} (2 - x)^{1/2} dx \]

2013

  1. \[ \int \log(x^2) - 2 \log(2x) dx \]

  2. \[ \int_{-1}^3 e^{|x|} dx \]

  3. \[ \int \frac{(\log x)(\cos x) - (\sin x)(1/x)}{(\log x)^2} dx \]

  4. \[ \int_1^{11} x^3 - 3x^2 + 3x - 1 dx \]

  5. \[ \int_0^2 \sqrt{12 - 3x^2} dx \]

  6. \[ \int_0^6 x + (x - 3)^7 + \sin(x - 3) dx \]

  7. \[ \int \sin x \sqrt{1 + \tan^2 x} dx \]

  8. \[ \int \frac{x^5 - x^3 + x^2 - 1}{x^4 - x^3 + x - 1} dx \]

  9. \[ \int_0^1 \log x dx \]

  10. \[ \int \frac{1}{1 - e^{-x}} dx \]

  11. \[ \int_0^\pi \sin^2 x \cos^2 x dx \]

  12. \[ \int_0^{441} \frac{\pi \sin(\pi \sqrt{x})}{\sqrt{x}} dx \]

  13. \[ \int \tan^2 x dx \]

  14. \[ \int_0^{256} (x - \lfloor x \rfloor)^2 dx \]

  15. \[ \int e^{\frac{4}{\sqrt{x}}} dx \]

  16. \[ \int \cos x \cot x dx \]

  17. \[ \int 2 \log x + (\log x)^2 dx \]

  18. \[ \int \frac{x^3}{1 + x^2} dx \]

  19. \[ \int \frac{1}{2 - 2x + x^2} dx \]

  20. \[ \int \sin x \log(\sin x) dx \]

  21. \[ \int \frac{x}{1 - x^4} dx \]

  22. \[ \int \sqrt{12 - 3x^2} dx \]

  23. \[ \int \sec^5 x \tan^3 x dx \]

  24. \[ \int_{-\pi/4}^{\pi/4} \frac{1}{1 - \sin x} dx \]

  25. \[ \int \frac{1}{x \sqrt{x^2 - 2}} dx \]

Integration Bees