[SOLVED] Calculus – Homework 7

Description

Problem 1

Use substitution to evaluate the following integrals.

1. (

Use integration by parts to evaluate the following integrals.

1. ^R cos(x)ln(sinx)dx 

Problem 2

(10 points)

1. Prove the reduction formula
   • points) Hint: Use integration by parts and the fact that cos²(x) + sin²(x) = 1.
2. 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:

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

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