Primitive of Square of Tangent Function

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Theorem

$\ds \int \tan^2 x \rd x = \tan x - x + C$

where $C$ is an arbitrary constant.


Proof 1

\(\ds \int \tan^2 x \rd x\) \(=\) \(\ds \int \paren {\sec^2 x - 1} \rd x\) Difference of Squares of Secant and Tangent
\(\ds \) \(=\) \(\ds \int \sec^2 x \rd x + \int \paren {-1} \rd x\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \tan x + C + \int \paren {-1} \rd x\) Primitive of Square of Secant Function
\(\ds \) \(=\) \(\ds \tan x - x + C\) Primitive of Constant

$\blacksquare$


Proof 2

\(\ds I_n\) \(=\) \(\ds \int \tan^n x \rd x\)
\(\ds \) \(=\) \(\ds \frac {\tan^{n - 1} x} {n - 1} - I_{n - 2}\) 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 $n = 2$

$\blacksquare$


Sources

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