if f is continuous on the closed interval [a,b] then there exixts a number c in the closed interval [a,b] such that:
b
Sigma f(x)dx = f(c)(b-a)
a
Second Fundamental Theorem of Calculus
When we defined the definite integral of f on the interval [a,b] we used the constant b as the upper limit of integration and x as the variable of integration. We now look at a slightly different situation in which the variable x is used as the upper limit of integration.
(d/dx)Sigma dx
Guidelines for Integration by Substitution
1. Choose a Substitution; choose the inner part of a composite function to sub
2. Compute du = g'(x)dx
3. Rewrite the integral in terms of the variable u
4. Evaluate the resulting integral in terms of u
5. Replace u by g(x) to obtain an antiderivative in terms of x