Question
Evaluate:
$\int\limits_0^\frac{\pi}{4} \log (1 + \tan\text{ x)dx}$
$\int\limits_0^\frac{\pi}{4} \log (1 + \tan\text{ x)dx}$
$\text{Using} $ $\int\limits_0^{a} \text f (x) {dx} = \int\limits_0^{a} f (a - x) \text{dx, we get}$
$\text {I} = \int\limits_0^\frac{\pi}{4} \log \bigg[1 + tan \bigg(\frac{x}{4} - {x}\bigg)\bigg] \text{dx}$
$\int\limits_0^\frac{\pi}{4} \log \bigg[1 + \frac{1 - \tan x}{1 + tan x}\bigg] \text{dx} = \int\limits_0^\frac{\pi}{4}[\log 2 - \log(1 + tan x)] \text{dx}$
$\therefore 2\text{I} = \bigg[\int\limits_0^\frac{\pi}{4} \log 2.\text{dx}\bigg] \Rightarrow \text{I} = \frac{\pi}{8} \log 2$
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$\tan\text{y}\frac{\text{dy}}{\text{dx}}=\sin(\text{x}+\text{y})+\sin(\text{x}-\text{y})$