Primitive of Power of Tangent of a x

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Theorem

$\ds \int \tan^n a x \rd x = \frac {\tan^{n - 1} a x} {\paren {n - 1} a} - \int \tan^{n - 2} a x \rd x$

for $n \ne 1$.


Proof

\(\ds \int \tan^n a x \rd x\) \(=\) \(\ds \int \tan^{n - 2} a x \tan^2 a x \rd x\)
\(\ds \) \(=\) \(\ds \int \tan^{n - 2} a x \paren {\sec^2 a x - 1} \rd x\) Difference of Squares of Secant and Tangent
\(\ds \) \(=\) \(\ds \int \tan^{n - 2} a x \sec^2 a x \rd x - \int \tan^{n - 2} \rd x\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac {\tan^{n - 1} a x} {\paren {n - 1} a} - \int \tan^{n - 2} \rd x\) Primitive of $\tan^n a x \sec^2 a x$

$\blacksquare$


Also see


Sources

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