Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$
Calculus – Integrate ?0 to ?/2 (1/(1+tan^?(x))) dx
calculusdefinite integralsintegrationtrigonometry
calculusdefinite integralsintegrationtrigonometry
Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$
Best Answer
Use the fact that
$$\tan{\left (\frac{\pi}{2}-x\right)} = \frac{1}{\tan{x}}$$
i.e.,
$$\frac1{1+\tan^{\alpha}{x}} = 1-\frac{\tan^{\alpha}{x}}{1+\tan^{\alpha}{x}} = 1-\frac1{1+\frac1{\tan^{\alpha}{x}}} = 1-\frac1{1+\tan^{\alpha}{\left (\frac{\pi}{2}-x\right)}}$$
Therefore, if the sought-after integral is $I$, then
$$I = \frac{\pi}{2}-I$$
and...