prove-that-I-0-pi-2-ln-1-sin-2-d-2G-pi-ln-2-G-catalan-constant-

Question Number 161089 by mnjuly1970 last updated on 12/Dec/21

p r o v e t h a t

I = \int_{0}^{\frac{π}{2}} \ln (1 + s i n (2 α)) d α

= 2 G - π \ln (\sqrt{2})

G : c a t a l a n c o n s t a n t

Answered by Ar Brandon last updated on 11/Dec/21

I = \int_{0}^{\frac{π}{2}} \ln (1 + \sin 2 α) d α

= \int_{0}^{\frac{π}{2}} \ln {(\cos α + \sin α)}^{2} d α

= 2 \int_{0}^{\frac{π}{2}} \ln (\cos α + \sin α) d α

= 2 (G - \frac{π \ln 2}{4}) = 2 G - \frac{π \ln 2}{2}

= 2 G - π \ln \sqrt{2}

Answered by mnjuly1970 last updated on 12/Dec/21

Commented by Tawa11 last updated on 12/Dec/21

Great sir

Answered by mnjuly1970 last updated on 12/Dec/21