integration tutorial: analysis

Consider the integral
int [ (2z + sin(2z)) / (z2 + sin(z)2) ] dz
This is an indefinite integral, so we need to find the most general antiderivative of the integrand f(z).
To find an antiderivative of f(z), we go through our list of integration methods:
  1. Recognize elementary antiderivatives
  2. Rewrite the integrand to make it easier
  3. Use substitution to reverse the chain rule or simplify the integrand
  4. Use integration by parts
  5. Use inspection to see the value of a definite integral
to find one that works for this integrand. In this case, only one method is appropriate. We use substitution to rewrite the integrand in terms of w = z2 + sin(z)2. This is shown below.
Substitution:
Let w = z2 + sin(z)2. Then w' = 2z + 2sin(z)cos(z), so dw = (2z + sin(2z)) dz. The integral can therefore be rewritten as
int [ (2z + sin(2z)) / (z2 + sin(z)2) ] dz = int 1/w dw = ln(w) + C
Thus, substituting back for w,
int [ (2z + sin(2z)) / (z2 + sin(z)2) ] dz = ln(z2 + sin(z)2) + C
[ ]
Tutorial links:
see a more detailed substitution example
see another substitution example like this one
do another practice gateway
integration analysis
Last Modified: Wed Feb 6 13:53:59 EST 2002
Comments to glarose@umich.edu
©2002 Gavin LaRose, UM Math Dept.