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, there are several methods that work. We can use integration by parts to rewrite the product of u = x and v' = cos(x) in the integrand using the integration by parts formula, or use inspection to determine the value of the integral. This is shown below: integration by parts; inspection.

Integration by parts:
Let u = x and v' = cos(x). Then u' = 1 and v = sin(x), so, remembering the integration by parts formula
we have To evaluate the definite integral, we take this antiderivative, evaluate it at the endpoints of the integral (-pi/2 and pi/2), and take the difference of the values. This gives

Inspection:
Note that we are integrating on odd function over an interval symmetric about the y-axis. Because the function is odd, the area to the left of the axis will be the negative of the area to the right, and the value of the integral must be 0.