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  4. If f(x)= tan x- tan 3x+ tan 5x-..... to ∞ with 0<x<(π /4), then ∫0(π /4)f(x)dx is equal to:

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Q. If $ f(x)=\tan x-{{\tan }^{3}}x+{{\tan }^{5}}x-..... $ to $ \infty $ with $ 0
KEAMKEAM 2006

Solution:

$ f(x)=\tan x-{{\tan }^{3}}x+{{\tan }^{5}}x-....\infty $ $ f(x)=\frac{\tan x}{1+{{\tan }^{2}}x}=\frac{\tan x}{{{\sec }^{2}}x}=\frac{\sin 2x}{2} $ $ \therefore $ $ \int_{0}^{\pi /4}{f(x)}dx=\int_{0}^{\pi /4}{\frac{\sin 2x}{2}} $ $ =\left[ -\frac{\cos 2x}{4} \right]_{0}^{\pi /4} $ $ =\frac{1}{4} $