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I came across a question in the text book where I had to prove,

$$ \int_0^1 \ln(\Gamma (x)) \, dx = \ln \sqrt{2π} $$

Here's my approach,

$$ I = \int_0^1 \ln(\Gamma (x)) \, dx $$

Also, using $$\int_a^b f(x) \, dx = \int_a^b f(a+b-x) \, dx$$ I get,

$$ I = \int_0^1 \ln \Gamma (1-x) \, dx $$

$$ 2I = \int_0^1 \ln ( \Gamma (x) \Gamma (1-x)) \, dx $$

$$ I = \frac{1}{2} \int_0^1 \ln \bigg[ \dfrac{π}{\sin π x} \bigg] \, dx $$

Now I can use the $ \ln (\frac{a}{b}) $ formula and solve it.

I was wondering, are there any other ways you can show me?

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