integration analysis

Consider the integral

ln(p) e^{p 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, only one method is appropriate. We use substitution to rewrite the integrand in terms of w = pix (we call this simple substitution "Ax" substitution). This is shown below.

Substitution (of "ax"):
Let w = p x. Then w' = p, so (1/(p)) dw = dx. The integral can therefore be rewritten as

ln(p) e^{p x} dx =

ln(p)e^w (1/(p)) dw = (ln(p)/p) e^w + C

Thus, substituting back for w,

ln(p) e^{p x} dx = (ln(p)/p) e^{p x} + C

integration tutorial: analysis