Use substitution to evaluate the following integrals.

Use integration by parts to evaluate the following integrals.

1. ^R cos(x)ln(sinx)dx

Problem 2

1. Prove the reduction formula

Hint: Use integration by parts and the fact that cos²(x) + sin²(x) = 1.

1. Suppose that f : R→R continuous and odd, i.e., satisfies −f(x) = f(−x). Show that

.

Problem 3

Using 2a)

Z

1. Evaluate cos²(x)dx.

Z

1. Evaluate cos³(x)dx.
2. Evaluate.

Bonus:

We will cover this on Tuesday, 26.11:

Find the area between the curves x = 1 − y² and y = −2x − 1.

Hint: At the end, integrate with respect to y not x.