nice-calculus-prove-that-0-pi-4-xcot-x-dx-1-2-G-pi-4-log-2-

Question Number 128245 by mnjuly1970 last updated on 05/Jan/21

n i c e c a l c u l u s \dots

p r o v e t h a t :: \int_{0}^{\frac{π}{4}} x c o t (x) d x = \frac{1}{2} (G + \frac{π}{4} l o g (2))

Answered by Dwaipayan Shikari last updated on 05/Jan/21

\int_{0}^{\frac{π}{4}} x c o t x d x = {[x l o g (s i n x)]}_{0}^{\frac{π}{4}} - \int_{0}^{\frac{π}{4}} l o g (s i n x) d x

= - \frac{π}{8} l o g (2) + \frac{π}{4} l o g (2) + \frac{G}{2} l o g (2) (H a s b e e n p r o v e d e a r l i a r)

= \frac{π}{8} l o g (2) + \frac{G}{2} l o g (2)

Q 128244

Commented by mnjuly1970 last updated on 05/Jan/21

g r a t e f u l m r p a y a n \dots m e r c e y \dots