integration tutorial: analysis

Consider the integral

This is an indefinite integral, so we need to find the most general antiderivative of the integrand f(y).

To find an antiderivative of f(y), 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 can rewrite the integrand: note that cos(-6y)=cos(6y) because cosine is an even function to give an expression that is more easily integrable, or can rewrite the integrand: break integrand into two terms to give an expression that is more easily integrable. This is shown below: rewriting; rewriting 1.

Rewriting:
Note that cos(-6y)=cos(6y) because cosine is an even function to rewrite the integral: Thus, which can be evaluated using inspection, to obtain

Rewriting 1:
Break integrand into two terms. This can be broken into separate integrals, which can be evaluated using substitution (of "ax"), and substitution (of "ax"), respectively, to obtain

Note that when there are multiple integration methods the antiderivatives may appear to be different but are actually the same function shifted vertically by different constants. This is reasonable because the most general antiderivative is only unique up to an additive constant.