1. Integration review

4. The Definite Integral

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

7. Substitution Rule

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

10. 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

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