integration tutorial: analysis

Consider the integral

This is a definite integral, so we need to find the area under the integrand f(x), which we do by using the Fundamental Theorem of Calculus: we find an antiderivative, evaluate it at the endpoints of the integral, and take the difference of the values.

To find an antiderivative of f(x), 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 = x². This is shown below.

Substitution:
Let w = x². Then w' = 2x, so (1/2) dw = x dx. The integral can therefore be rewritten as Thus, substituting back for w,
To evaluate the definite integral, we take this antiderivative, evaluate it at the endpoints of the integral (0 and 3), and take the difference of the values. This gives