To ensure success in this course, your integration skills must be flawless. Practice with the following exercises, and discuss among yourselves the best techniques to evaluate the integrals below. Feel free to drop questions and comments, and I will try to guide you in the right direction — without offering the solution, of course; that will spoil the fun for others that want to try by themselves.

1. \( \displaystyle{\int_0^3} \dfrac{dx}{x-1} \)
2. \( \displaystyle{\int_{-\infty}^0} xe^{-x}\, dx \)
3. \( \displaystyle{\int} \cos x \big( 1 + \sin^2 x \big)\, dx \)
4. \( \displaystyle{\int} \dfrac{\sin x + \sec x}{\tan x}\, dx \)
5. \( \displaystyle{\int_1^3} r^4 \ln r\, dr \)
6. \( \displaystyle{\int} \dfrac{x-1}{x^2-4x+5}\, dx \)
7. \( \displaystyle{\int} \sin^3 \theta \cos^5 \theta\, d\theta \)
8. \( \displaystyle{\int} x \sin^2 x\, dx \)
9. \( \displaystyle{\int} e^{x+e^x}\, dx \)
10. \( \displaystyle{\int} \dfrac{\ln x}{x \sqrt{1+ \big( \ln x \big)^2}}\, dx \)
11. \( \displaystyle{\int} \big( 1+\sqrt{x} \big)^8 \, dx \)
12. \( \displaystyle{\int} \ln \big( x^2 - 1 \big) \, dx \)
13. \( \displaystyle{\int_0^3} \dfrac{dx}{x-1} \)
14. \( \displaystyle{\int} \dfrac{3x^2-2}{x^2-2x-8}\, dx \)
15. \( \displaystyle{\int} \dfrac{dx}{1+e^x} \)
16. \( \displaystyle{\int} \sqrt{3-2x-x^2}\, dx \)
17. \( \displaystyle{\int} \dfrac{1+\cot x}{4-\cot x}\, dx \)
18. \( \displaystyle{\int} \sin(4x)\cos(3x)\, dx \)
19. \( \displaystyle{\int} e^x \sqrt{1+e^x}\, dx \)
20. \( \displaystyle{\int} \sqrt{1+e^x}\, dx \)
21. \( \displaystyle{\int} x^5 e^{-x^3}\, dx \)
22. \( \displaystyle{\int} \dfrac{1+\sin x}{1-\sin x}\, dx \)
23. \( \displaystyle{\int} \dfrac{dx}{3-5\sin x} \)
24. \( \displaystyle{\int} \dfrac{dx}{3\sin x - 4\cos x}\)