Primitive of Square of Tangent Function/Proof 2
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Theorem
- $\ds \int \tan^2 x \rd x = \tan x - x + C$
Proof
| \(\ds I_n\) | \(=\) | \(\ds \int \tan^n x \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\tan^{n - 1} x} {n - 1} - I_{n - 2}\) | Reduction Formula for Integral of Power of Tangent | |||||||||||
| \(\ds I_0\) | \(=\) | \(\ds \int \paren {\tan x}^0 \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \d x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds x + C\) | Primitive of Constant | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds I_2\) | \(=\) | \(\ds \tan x - x + C'\) | setting $n = 2$ |
$\blacksquare$