calculate-1-3-1-2-x-1-x-dx-with-x-0-t-x-1-e-t-dt-with-x-gt-0-

Question Number 53950 by maxmathsup by imad last updated on 27/Jan/19

c a l c u l a t e \int_{\frac{1}{3}}^{\frac{1}{2}} Γ (x) Γ (1 - x) d x w i t h Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t w i t h x > 0 .

Commented by maxmathsup by imad last updated on 30/Jan/19

w e h a v e f o r 0 < x < 1 Γ (x) . Γ (1 - x) = \frac{π}{s i n (π x)} (c o m p l m e n t s f o r m u l a) \Rightarrow

\int_{\frac{1}{3}}^{\frac{1}{2}} Γ (x) . Γ (1 - x) d x = π \int_{\frac{1}{3}}^{\frac{1}{2}} \frac{d x}{s i n (π x)} =_{π x = t} π \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{d t}{π s i n (t)}

= \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{d t}{s i n t} =_{t a n (\frac{t}{2}) = u} \int_{\frac{1}{\sqrt{3}}}^{1} \frac{1}{\frac{2 u}{1 + u^{2}}} \frac{2 d u}{1 + u^{2}} = \int_{\frac{1}{\sqrt{3}}}^{1} \frac{d u}{u} = {[l n ∣ u ∣]}_{\frac{1}{\sqrt{3}}}^{1} = - l n (\frac{1}{\sqrt{3}})

= l n (\sqrt{3}) .