0-pi-2-x-sin-2-x-ln-sin-x-dx-

Question Number 155252 by mnjuly1970 last updated on 27/Sep/21

\int_{0}^{\frac{π}{2}} x . {s i n}^{2} (x) . l n (s i n (x)) d x

Answered by phanphuoc last updated on 28/Sep/21

l n s i n x = - l n 2 - \sum_{k = 1}^{\infty} \frac{c o s (2 k x)}{k}

\to I = \int_{0}^{\infty} {x s i n}^{2} x (- l n 2 - Σ \frac{c o s (2 k x)}{k}) d x

= - l n 2 \int_{0}^{\infty} {x s i n}^{2} x d x - Σ \frac{1}{2 k} x (1 - c o s 2 x) c o s (2 k x)

y o u c a n I B P \to \dots \dots