Mathematics 136 – Advanced Placement Calculus Practice Problems on Integrals by Substitution October 16, 2009

Integrate each of the following by an appropriate substitution, and check your work by differentiating your answer. (Turn over for problem 5.)

1. ∫ sin^{7}(2x) cos(2x) dx.

2. ∫ \frac{tan^{−1}(x)}{x^{2} + 1} dx.

3. ∫ \frac{x^{2} + 3}{x^{3} + 9x + 14} dx.

4. ∫ \frac{e^{\sqrt{x}}}{\sqrt{x}} dx.

5. ∫ x(x + 3)^{3/5} dx. (Let u = x + 3 and convert everything to the new variable u.) 2