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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?

StubbornAtom
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William
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  • @Robert Z Yes! Corrected. It was a typing error. Thanks. – William Jun 09 '18 at 10:38
  • Then for the last integral see https://math.stackexchange.com/questions/354795/evaluate-int-0-pi-ln-left-sin-theta-right-d-theta – Robert Z Jun 09 '18 at 10:42
  • See also https://math.stackexchange.com/questions/167206/calculate-integrals-involving-gamma-function/167215#167215 and https://math.stackexchange.com/questions/529205/evaluating-the-log-gamma-integral-int-0z-log-gamma-x-mathrm-dx-i – Robert Z Jun 09 '18 at 10:53
  • @Robert Z how are you able to find the relevant questions like that? What do you type in the search? – William Jun 09 '18 at 10:55
  • I use https://approach0.xyz/search/ – Robert Z Jun 09 '18 at 10:57

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