Question
Using properties of definite integrals, evaluate:
$\int\limits^{\pi/4}_{0}\text{log (1 + tan x) dx}$.
$\int\limits^{\pi/4}_{0}\text{log (1 + tan x) dx}$.
$=\int_{0}^{\pi/4}\log\Bigg(1+\frac{1-\tan\text{x}}{\text{1 + tan x}}\Bigg)\text{dx}=\int_{0}^{\pi/4}\log\Bigg(\frac{2}{\text{1 + tan x}}\Bigg)\text{dx}$
$=\int_{0}^{\pi/4}\log2\text{ dx}-\int_0^{\pi/4}\log(1+\tan\text{ x})\text{ dx}$
$\Rightarrow\text{2I}=\log2\cdot\int_0^{\pi/4}1\cdot\text{dx}=\log2\cdot\pi/4$
$\Rightarrow\text{I}=\pi/8\cdot\log2.$
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$\frac{\text{dy}}{\text{dx}} - \frac{\text{y}}{\text{x}} + \text{cosec} \bigg(\frac{\text{y}}{\text{x}}\bigg) = \text {0; y = 0 when x} = 1.$
$(\text{x + y})\frac{\text{dy}}{\text{dx}}=1$