integration tutorial: analysis

Consider the integral

This is a definite integral, so we need to find the area under the integrand f(t), 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(t), 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 basic antidifferentiation to determine the antiderivative, or use inspection to determine the value of the integral. This is shown below: basic antidifferentiation; inspection.

Basic antidifferentiation:
In this case, by using rules for basic antiderivatives we can just write down the antiderivative of the integrand:
To evaluate the definite integral, we take this antiderivative, evaluate it at the endpoints of the integral (0 and 2pi), and take the difference of the values. This gives

Inspection:
Note that we are integrating the cosine function over (an integer multiple of) its period. Because of the periodicity, the area above and below the x-axis will be equal and the value of the integral must be 0.