MATH114 Chapter Notes - Chapter 1.5: Logarithm, Antiderivative, Inverse Function

• School: University of Alberta
• Department: Mathematics
• Course: MATH114
• Professor: Vincent Bouchard

Document Summary

The substitution rule: if u = g(x) is a di erentiable function whose range is an interval. I, and if f is continuous on i, then(cid:90) f (g(x))g(cid:48)(x)dx = f (u)du. (cid:90) (cid:90) g(b) In practice, to evaluate an integral using substitution one nds an appropriate substitution u = g(x); then du = g(cid:48)(x)dx, and one rewrites the integral in terms of u instead of x. If the substitution was good, the resulting integral in u is simpler and an antiderivative can be found explicitly. Substitution and the de nite integral: if g(cid:48) is continuous on [a, b] and f is continuous on the range of g(x) = u, then(cid:90) b f (g(x)) g(cid:48)(x)dx = f (u)du. a g(a) What this is saying is that to evaluate a de nite integral using a substitution u = g(x), we must also change the limits of integration from the variable x to the variable u.