# Primitive of Exponential of a x over Power of x

## Theorem

$\displaystyle \int \frac {e^{a x} \rd x} {x^n} = \frac {-e^{a x} } {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {e^{a x} \rd x} {x^{n - 1} } + C$

where $n \ne 1$.

## Proof

With a view to expressing the primitive in the form:

$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

 $\displaystyle u$ $=$ $\displaystyle e^{a x}$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {\d u} {\d x}$ $=$ $\displaystyle a e^{a x}$ Derivative of $e^{a x}$

and let:

 $\displaystyle \frac {\d v} {\d x}$ $=$ $\displaystyle \frac 1 {x^n}$ $\displaystyle$ $=$ $\displaystyle x^{-n}$ $\displaystyle \leadsto \ \$ $\displaystyle v$ $=$ $\displaystyle \frac {x^{-n + 1} } {-n + 1}$ Primitive of Power $\displaystyle$ $=$ $\displaystyle \frac {-1} {\paren {n - 1} x^{n - 1} }$ simplifying

Then:

 $\displaystyle \int \frac {e^{a x} \rd x} {x^n}$ $=$ $\displaystyle e^{a x} \paren {\frac {-1} {\paren {n - 1} x^{n - 1} } } - \int \paren {\frac {-1} {\paren {n - 1} x^{n - 1} } } \paren {a e^{a x} } \rd x + C$ Integration by Parts $\displaystyle$ $=$ $\displaystyle \frac {-e^{a x} } {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {e^{a x} \rd x} {x^{n - 1} } + C$ Primitive of Constant Multiple of Function

$\blacksquare$