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Q.
The value of definite integral $\int\limits_0^{\frac{\pi}{12}} \frac{\tan ^2 x-3}{3 \tan ^2 x-1} dx$ is equal to
Integrals
Solution:
Put $\tan x=t \Rightarrow \sec ^2 x d x=d t$
So $I=\int\limits_0^{\frac{\pi}{12}} \frac{\tan ^2 x-3}{3 \tan ^2 x-1} d x=\int\limits_0^{2-\sqrt{3}} \frac{t^2-3}{\left(3 t^2-1\right)\left(1+t^2\right)} d t$
$=\int\limits_0^{2-\sqrt{3}}\left(\frac{1}{1+ t ^2}-\frac{2}{3 t ^2-1}\right) dt =\int\limits_0^{2-\sqrt{3}} \frac{ dt }{1+ t ^2}-\frac{2}{3} \int\limits_0^{2-\sqrt{3}} \frac{ dt }{ t ^2-\left(\frac{1}{\sqrt{3}}\right)^2}$
$=\frac{\pi}{12}+\frac{1}{\sqrt{3}} \ln \left(\frac{\sqrt{3}-1}{2-\sqrt{3}}\right)$