advanced-calculus-evaluate-Im-0-pi-2-li-2-sin-x-li-2-csc-x-dx-

Question Number 136060 by mnjuly1970 last updated on 18/Mar/21

\dots a d v a n c e d c a l c u l u s \dots .

e v a l u a t e ::

ϕ = Im (\int_{0}^{\frac{π}{2}} {l i}_{2} (s i n (x)) + {l i}_{2} (c s c (x)) d x)

Answered by mindispower last updated on 18/Mar/21

c s c (x) = \frac{1}{s i n (x)}

w e h a v e

{l i}_{2} (z) + {l i}_{2} (\frac{1}{z}) = - \frac{π^{2}}{6} - \frac{{l n}^{2} (- z)}{2}

\Leftrightarrow ϕ = I m \int_{0}^{\frac{π}{2}} (- \frac{π^{2}}{6} - \frac{{l n}^{2} (- s i n (x))}{2}) d x

= - \frac{I m}{2} \int_{0}^{\frac{π}{2}} {(l n (s i n (x)) + i π)}^{2} d x

= - \frac{I m}{2} \int_{0}^{\frac{π}{2}} (2 i π l n (s i n (x))) d x

= - π \int_{0}^{\frac{π}{2}} l n (s i n (x)) d x = \frac{π^{2} l o g (2)}{2}

Commented by mnjuly1970 last updated on 18/Mar/21

t h a n k s a l o t s i r p o w e r \dots

Commented by mindispower last updated on 19/Mar/21

p l e a s u r s i r