integration analysis

Consider the integral

sec(x)tan(x) dx

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

To find an antiderivative of f(x), we go through our list of integration methods:

Recognize elementary antiderivatives
Rewrite the integrand to make it easier
Use substitution to reverse the chain rule or simplify the integrand
Use integration by parts
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: use the definitions of tangent and secant to give an expression that is more easily integrable, or use basic antidifferentiation to determine the antiderivative. This is shown below: rewriting; basic antidifferentiation.

Rewriting:
Use the definitions of tangent and secant to rewrite the integral:

sec(x)tan(x) = [ sin(x) / cos(x)² ].

Thus,

sec(x)tan(x) dx =

sin(x) / cos(x)²dx

which can be evaluated using substitution, to obtain

sin(x) / cos(x)²dx = (-(cos(x))^-1) + C.

Explanation for rewritten term(s)
Substitution
Let w = cos(x). Then w' = -sin(x), so (-1) dw = sin(x) dx. The integral can therefore be rewritten as

sin(x) / cos(x)² dx =

w^-2 (-1) dw = -w^-1 + C

Thus, substituting back for w,

sin(x) / cos(x)² dx = -(cos(x))^-1 + C

Basic antidifferentiation:
In this case, by using rules for basic antiderivatives we can just write down the antiderivative of the integrand:

sec(x)tan(x) dx = sec(x) + C

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.

integration tutorial: analysis