Primitive of Reciprocal of Cube of Cosine of a x

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Theorem

$\displaystyle \int \frac {\d x} {\cos^3 a x} = \frac {\sin a x} {2 a \cos^2 a x} + \frac 1 {2 a} \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$


Proof

\(\displaystyle \int \frac {\d x} {\cos^3 a x}\) \(=\) \(\displaystyle \int \sec^3 a x \rd x\) Secant is Reciprocal of Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sec a x \tan a x} {2 a} + \frac 1 2 \int \sec a x \rd x\) Primitive of $\sec^n a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sin a x} {2 a \cos^2 a x} + \frac 1 2 \int \sec a x \rd x\) Secant is $\dfrac 1 \cos$ and Tangent is $\dfrac \sin \cos$
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sin a x} {2 a \cos^2 a x} + \frac 1 {2 a} \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C\) Primitive of $\sec a x$

$\blacksquare$


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