MATH 132, section 1 – Calculus for Physical and Life Sciences Discussion 1 – Integration By Substitution Practice January 30, 2008

Each of the following integrals can be done by the method of u-substitution: 1. Determine the appropriate function u,

2. Compute du,

3. Change the integral to an equivalent form in the new variable u,

4. Integrate in terms of u, then

5. Resubstitute u to express the answer in terms of the original variable. Do all the integrals below, and put the work for your group’s integral up on the board.

(I) ∫ (x + 3)(x^{2} + 6x + 12)^{5/4} dx

(II) ∫ sin(1 + sin θ) cos θ dθ

(III) ∫ cos^{3}(4θ) sin(4θ) dθ

(IV) ∫ \frac{1}{\sqrt{1 − 4x^{2}}} dx

(V) ∫ \frac{x}{\sqrt{1 − 4x^{2}}} dx

(VI) ∫ \frac{1}{x ln(x)} dx

(VII) ∫ \frac{cos(\sqrt{y})}{\sqrt{y}} dy.

(VIII) ∫ \frac{x}{\sqrt{x + 1}} dx (Hint: Try u = x + 1.)

(IX) ∫ cot θ dθ (Hint: Rewrite cot θ in terms of sin θ and cos θ to start.)