Reduction Formula for Integral of Power of Tangent
Jump to navigation
Jump to search
Theorem
For all $n \in \Z_{> 1}$:
- $\ds \int \map {\tan^n} x \rd x = \frac {\map {\tan^{n - 1} } x} {n - 1} - \int \map {\tan^{n - 2} } x \rd x$
Proof
| \(\ds \int \map {\tan^n} x \rd x\) | \(=\) | \(\ds \int \map {\tan^{n - 2} } x \map {\tan^2} x \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \map {\tan^{n - 2} } x \paren {\map {\sec^2} x - 1} \rd x\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \map {\tan^{n - 2} } x \map {\sec^2} x \rd x - \int \map {\tan^{n - 2} } x \rd x\) | Linear Combination of Primitives |
Let:
| \(\ds t\) | \(=\) | \(\ds \map \tan x\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {\d t} {\d x}\) | \(=\) | \(\ds \map {\sec^2} x\) | Derivative of Tangent Function |
Then:
| \(\ds \int \map {\tan^{n - 2} } x \map {\sec^2} x \rd x - \int \map {\tan^{n - 2} } x \rd x\) | \(=\) | \(\ds \int t^{n - 2} \rd t - \int \map {\tan^{n - 2} } x \rd x\) | Integration by Substitution | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {t^{n - 1} } {n - 1} - \int \map {\tan^{n - 2} } x \rd x\) | Primitive of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\map {\tan^{n - 1} } x} {n - 1} - \int \map {\tan^{n - 2} } x \rd x\) | substituting back $t \to \map \tan x$ |
$\blacksquare$