Indefinite Integration 1 Question 71

72. Integrate $\int _0^{\pi / 4} \log (1+\tan x) d x$.

(1997C, 2M)

Show Answer

Solution:

  1. Let $I=\int _0^{\pi / 4} \log (1+\tan x) d x$

$$ \begin{aligned} & I=\int _0^{\pi / 4} \log \left(1+\tan \left(\frac{\pi}{4}-x\right)\right) d x \\ \therefore \quad & I=\int _0^{\pi / 4} \log 1+\frac{1-\tan x}{1+\tan x} d x \end{aligned} $$

$$ \begin{aligned} & \qquad \int _0^{\pi / 4} \log \frac{1+\tan x+1-\tan x}{1+\tan x} d x \\ & \qquad I=\int _0^{\pi / 4} \log \frac{2}{1+\tan x} d x \Rightarrow I=\int _0^{\pi / 4} \log 2 d x-I \\ & \Rightarrow \quad 2 I=\frac{\pi}{4} \log 2 \Rightarrow \quad I=\frac{\pi}{8}(\log 2) \\ & \text {



NCERT Chapter Video Solution

Dual Pane