Integration by parts (IBP)  Lecture Video

Evaluate the following integrals:

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

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

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

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

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

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

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

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

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

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

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

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

13. \(\int x^3\, e^{x^2}\, \mathrm{d}x\)  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