calculate-0-1-0-1-x-t-2-x-t-1-x-dx-dt-m-n-

Question Number 159447 by mnjuly1970 last updated on 17/Nov/21

You can't use 'macro parameter character #' in math mode

Ω := \int_{0}^{1} \int_{0}^{1} \frac{x^{\frac{t}{2}} - x^{t}}{1 - x} d x d t = ?

- - - m . n - - -

Commented by Ar Brandon last updated on 17/Nov/21

Answered by Ar Brandon last updated on 17/Nov/21

Ω = \int_{0}^{1} \int_{0}^{1} \frac{x^{\frac{t}{2}} - x^{t}}{1 - x} d x d t

= \int_{0}^{1} (ψ (t + 1) - ψ (\frac{t}{2} + 1)) d t

= {[\ln (Γ (t + 1)) - 2 \ln (Γ (\frac{t}{2} + 1))]}_{0}^{1}

= \ln (\frac{Γ (2)}{Γ^{2} (\frac{3}{2})}) - \ln (\frac{Γ (1)}{Γ^{2} (1)}) = \ln (\frac{4}{π})

Commented by mnjuly1970 last updated on 17/Nov/21

p e a c e b e u p o n y o u s i r b r a n d o n . .

Commented by Ar Brandon last updated on 17/Nov/21

With you too, Sir.