Primitive of Square of Tangent of a x

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Theorem

$\ds \int \tan^2 a x \rd x = \frac {\tan a x} a - x + C$


Proof

\(\ds \int \tan^2 x \rd x\) \(=\) \(\ds \tan x - x + C\) Primitive of $\tan^2 x$
\(\ds \leadsto \ \ \) \(\ds \int \tan^2 a x \rd x\) \(=\) \(\ds \frac 1 a \paren {\tan a x - a x} + C\) Primitive of Function of Constant Multiple
\(\ds \) \(=\) \(\ds \frac {\tan a x} a - x + C\) simplifying

$\blacksquare$


Also see


Sources