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  4. If ∫ limits01 (λ n x /√1- x 2) dx = k ∫0π ln (1+ cos x ) dx then the value of k is :

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Q. If $\int\limits_0^1 \frac{\lambda n x }{\sqrt{1- x ^2}} dx = k \int_0^\pi \ln (1+\cos x ) dx$ then the value of $k$ is :

Integrals

Solution:

Put $x=\sin \theta \Rightarrow$ L HS $=\int\limits_0^{\pi / 2} \ln (\sin \theta) d \theta=k \int\limits_0^\pi \ln (1-\cos x)$
$=2 k \int\limits_0^{\pi / 2} \ln (\sin x) d x \Rightarrow k=1 / 2$