Integration by parts (IBP)

Evaluate the following integrals:

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

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

  3. \((i) \displaystyle \int y^7 \ln y \, \mathrm{d}y \qquad (ii) \int x^5 \ln (2x) \, \mathrm{d}x \qquad (iii) \int x^2 \sin(x) \, \mathrm{d}x\)   solution

  4. \((i) \displaystyle \int x\,e^{2x}\, \mathrm{d}x \qquad (ii) \int x^2\,e^{2x}\, \mathrm{d}x \qquad (iii) \int x^3\,e^{2x}\, \mathrm{d}x \)   solution

  5. \( (i) \displaystyle \int x \cos (x)\ \mathrm{d}x \qquad (ii) \int 5te^{-5t} \, \mathrm{d}t \)   solution

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

  7. \(\displaystyle (i) \int (x^2+1) \cos(2x) \, \mathrm{d}x \qquad (ii) \int (3x-5) e^{x} \, \mathrm{d}x \)  solution

  8. \(\displaystyle (i) \int x e^{5x} \, \mathrm{d}x \qquad (ii) \int x^2 e^{-x} \, \mathrm{d}x \)   solution

  9. \((i) \displaystyle \int 5x \cos (3x)\ \mathrm{d}x \qquad (ii) \int (5x+2)\sin \left(\frac{x}{4} \right) \ \mathrm{d}x \)  solution

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

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

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

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

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

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

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

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

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

  19. \((i) \displaystyle \int_{0}^1 (x^2+1) \, e^x\ \mathrm{d}x \qquad (ii) \int_{0}^{\pi/4} 2x \, \sin(2x)\ \mathrm{d}x \)   solution

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