Primitive of Cube of Tangent of a x

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Theorem

$\ds \int \tan^3 a x \rd x = \frac {\tan^2 a x} {2 a} + \frac 1 a \ln \size {\cos a x} + C$


Proof 1

\(\ds \int \tan^3 a x \rd x\) \(=\) \(\ds \int \tan a x \tan^2 a x \rd x\)
\(\ds \) \(=\) \(\ds \int \tan a x \paren {\sec^2 a x - 1} \rd x\) Difference of Squares of Secant and Tangent
\(\ds \) \(=\) \(\ds \int \tan a x \sec^2 a x \rd x - \int \tan a x \rd x\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac {\tan^2 a x} {2 a} - \int \tan a x \rd x + C\) Primitive of $\tan^n a x \sec^2 a x$: $n = 1$
\(\ds \) \(=\) \(\ds \frac {\tan^2 a x} {2 a} - \paren {\frac {-\ln \size {\cos a x} } a} + C\) Primitive of $\tan a x$
\(\ds \) \(=\) \(\ds \frac {\tan^2 a x} {2 a} + \frac 1 a \ln \size {\cos a x} + C\) simplifying

$\blacksquare$


Proof 2

\(\ds I_n\) \(=\) \(\ds \int \map {\tan^n} {a x} \rd x\)
\(\ds \) \(=\) \(\ds \frac {\map {\tan^{n - 1} } {a x} } {a \paren {n - 1} } - I_{n - 2}\) Reduction Formula for Integral of Power of Tangent
\(\ds I_1\) \(=\) \(\ds -\frac 1 a \ln \size {\map \cos {a x} } + C\) Primitive of $\tan a x$: Cosine Form
\(\ds \leadsto \ \ \) \(\ds I_3\) \(=\) \(\ds \frac {\map {\tan^2} {a x} } {2 a} + \frac 1 a \ln \size {\map \cos {a x} } + C'\)

$\blacksquare$


Also see


Sources

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