Integration by parts (IBP)  Lecture Video

Evaluate the following integrals:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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