14.4 Problems

Problem 1. Use integration by parts to find the following antiderivates.

(a) ∫ sin(x)cos(x)dx (b) ∫ xe^xdx (c) ∫ x²e^xdx (d) ∫ e^x sinxdx

Problem 2. Use integration by substitution to find the following antiderivatives.

(a) ∫ xcos(x²)dx (b) ∫ sin(x)e^{-x2}dx (c) ∫ (d) ∫ xdx

Problem 3. Let f : [a,b] →ℝ be an integrable function. Show that

Problem 4. Let f,g : [a,b] →ℝ be two integrable functions such that jfj² and jgj² are integrable as well. Show that

Hint: revisit Chapter 2 about normed spaces, and find an inequality that feels similar to this one.

Problem 5. The famous Dirichlet function is defined by

Is D(x) integrable?