Integration review

The Definite Integral  

Net Area on interval [-1,9]? Total Area on interval [-1,9]?

Substitution Rule

Example: Evaluate:  x3 cos(x4 + 2) dx

Solutions: 1) We make the substitution u = x4 + 2 because its differential is du = 4x3 dx, which, apart from the constant factor 4, occurs in the integral. Thus, using x3 dx =(1/4)du and the Substitution Rule, we have  x3 cos(x4 + 2) dx = (1/4)  cos u  du = (1/4)  cos u du = (1/4) sin u + C = (1/4) sin(x4 + 2) + C

2) Let u = 2x + 1. Then du = 2 dx, so dx = (1/2) du. (complete this blank!) 4