Integration by parts (IBP):   Lecture Video

Evaluate the following integrals:

  1. \( \displaystyle \int \ln (x) \ \mathrm{d}x \qquad \quad \int x \ln (x) \ \mathrm{d}x \) solution

  2. \( \displaystyle \int x^7 \ln (x) \ \mathrm{d}x \) solution

  3. \( \displaystyle \int x\,e^{2x}\, \mathrm{d}x \qquad \quad \int x^2\,e^{2x}\, \mathrm{d}x \qquad \quad \int x^3\,e^{2x}\, \mathrm{d}x \) solution

  4. \( \displaystyle \int 5te^{-5t} \, \mathrm{d}t \) solution

  5. \( \displaystyle \int \ln (\sqrt{x}) \, \mathrm{d}x \) solution

  6. \( \displaystyle \int x \cos (x)\ \mathrm{d}x \) solution

  7. \( \displaystyle \int 5x \cos (3x)\ \mathrm{d}x \) solution

  8. \(\displaystyle \int (5x+2)\sin \left(\frac{x}{4} \right) \ \mathrm{d}x \) solution

  9. \( \displaystyle \int x^2 \cos (2x)\ \mathrm{d}x \qquad \int x^3 \cos (2x)\ \mathrm{d}x \) solution

  10. \( \displaystyle \int x^2 \sin (x)\ \mathrm{d}x \) solution

  11. \( \displaystyle \int e^x \cos (x) \ \mathrm{d}x \) solution

  12. \( \displaystyle \int 6 \sin (3y) \, e^{2y} \ \mathrm{d}y \) solution

  13. \( \displaystyle \int \arcsin (x)\ \mathrm{d}x \) solution

  14. \( \displaystyle \int x^3\, e^{x^2}\, \mathrm{d}x\)solution

  15. \(\displaystyle \int \tan^{-1} (2x)\ \mathrm{d}x \) solution

  16. \(\displaystyle{\int_1^{\sqrt{3}} 8\arctan(1/x)\ \mathrm{d}x }\)solution

  17. \(\displaystyle{\int (\ln (x))^2 \mathrm{d}x }\) solution

  18. \(\displaystyle{\int_{0}^1 (x^2+1) \, e^x\ \mathrm{d}x}\) solution

  19. \(\displaystyle{\int \sin \sqrt{x} \, \mathrm{d}x }\) solution