Question

Prove that $\dfrac12 \int\limits_0^1 \log(\sin \pi x) dx = \dfrac{1}{2\pi} \int\limits_0^\pi \log(\sin x) dx$

Answer 1

$\large\underline{\sf{Solution-}}$

Consider, the integral

$\rm :\longmapsto\:\dfrac12 \: \displaystyle\int\limits_0^1 \log(\sin \pi x) dx$

To evaluate this integral, we use Method of Substitution

So, Substitute

$\purple{\rm :\longmapsto\:\pi \: x \: = \: y}$

$\purple{\rm :\longmapsto\:\pi \: dx \: = \: dy}$

$\purple{\rm :\longmapsto\: dx \: = \: \dfrac{dy}{\pi} }$

Now, We know, In definite integrals, when we substitute, we have to change the limits too. So,

$\purple{\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf 0 \\ \\ \sf 1 & \sf \pi \end{array}} \\ \end{gathered}}$

So, on substituting these, we get

$\rm \:  =  \: \:\dfrac{1}{2\pi} \: \displaystyle\int\limits_0^\pi \log(\sin y) dy$

We know,

$\purple{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\int\limits_a^bf(x)dx = \displaystyle\int\limits_a^bf(y)dy}}}$

So, using this, we get

$\rm \:  =  \: \:\dfrac{1}{2\pi} \: \displaystyle\int\limits_0^\pi \log(\sin x) dx$

Hence,

$\boxed{\tt{ \rm \:\dfrac12 \displaystyle\int\limits_0^1 \log(\sin \pi x) dx  =  \: \:\dfrac{1}{2\pi} \: \displaystyle\int\limits_0^\pi \log(\sin x) dx}}$

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MORE TO KNOW

$\purple{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\int\limits_a^bf(x)dx = \displaystyle\int\limits_a^bf(a + b - x)dx}}}$

$\purple{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\int\limits_0^af(x)dx = \displaystyle\int\limits_0^af(a - x)dx}}}$

$\purple{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\int\limits_{ - a}^af(x)dx = 2 \displaystyle\int\limits_0^af(x)dx \: if \: f( - x) = f(x)}}}$

$\purple{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\int\limits_{ - a}^af(x)dx = 0 \: if \: f( - x) = - f(x)}}}$

$\purple{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\int\limits_0^{2a}f(x)dx = 2 \displaystyle\int\limits_0^af(x)dx \: if \: f(2a - x) = f(x)}}}$

$\purple{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\int\limits_0^{2a}f(x)dx = 0 \: if \: f(2a - x) = - f(x)}}}$