Definite Integral from 0 to Half Pi of Reciprocal of One plus Power of Tan x

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Theorem

$\ds \int_0^{\pi/2} \frac {\d x} {1 + \tan^m x} = \frac \pi 4$

where $m$ is a real number.


Proof

\(\ds \int_0^{\pi/2} \frac {\d x} {1 + \tan^m x}\) \(=\) \(\ds \int_0^{\pi/2} \frac {\cos^m x} {\cos^m x + \sin^m x} \rd x\) multiplying by $\dfrac {\cos^m x} {\cos^m x}$
\(\ds \) \(=\) \(\ds \int_0^{\pi/2} \frac {\map {\cos^m} {\frac \pi 2 - x} } {\map {\cos^m} {\frac \pi 2 - x} + \map {\sin^m} {\frac \pi 2 - x} } \rd x\) Integral between Limits is Independent of Direction
\(\ds \) \(=\) \(\ds \int_0^{\pi/2} \frac {\sin^m x} {\sin^m x + \cos^m x} \rd x\) Cosine of Complement equals Sine, Sine of Complement equals Cosine

So:

\(\ds 2 \int_0^{\pi/2} \frac {\d x} {1 + \tan^m x}\) \(=\) \(\ds \int_0^{\pi/2} \frac {\sin^m x + \cos^m x} {\sin^m x + \cos^m x} \rd x\)
\(\ds \) \(=\) \(\ds \int_0^{\pi/2} \rd x\)
\(\ds \) \(=\) \(\ds \frac \pi 2\) Primitive of Constant

giving:

$\ds \int_0^{\pi/2} \frac {\d x} {1 + \tan^m x} = \frac \pi 4$

$\blacksquare$


Sources

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