integration tutorial: analysis

Consider the integral

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

To find an antiderivative of f(r), 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 substitution to rewrite the integrand in terms of w = e^r, or can rewrite the integrand: use properties of the exponents to combine numerator and denominator of the fraction to give an expression that is more easily integrable. This is shown below: substitution; rewriting.

Substitution:
Let w = e^r. Then w' = e^r, so dw = e^r dr. The integral can therefore be rewritten as Thus, substituting back for w,

Rewriting:
Use properties of the exponents to combine numerator and denominator of the fraction to rewrite the integral: Thus, which can be evaluated using substitution (of "ax"), 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.