let-0-lt-a-lt-1-calculate-0-ln-t-t-a-1-1-t-dt-and-0-ln-2-t-t-a-1-1-t-dt-

Question Number 73238 by mathmax by abdo last updated on 08/Nov/19

l e t 0 < a < 1 c a l c u l a t e \int_{0}^{\infty} \frac{l n (t) t^{a - 1}}{1 + t} d t a n d \int_{0}^{\infty} \frac{{l n}^{2} (t) t^{a - 1}}{1 + t} d t

Commented by mathmax by abdo last updated on 10/Nov/19

w e h a v e \int_{0}^{\infty} \frac{t^{a - 1}}{1 + t} d t = \frac{π}{s i n (π a)} = f (a) i f 0 < a < 1 (t h i s r e s u l t i s p r o v e d i n

t h e p l s t f o r m) \Rightarrow f^{^{'}} (a) = \int_{0}^{\infty} \frac{\partial}{\partial a} (\frac{t^{a - 1}}{1 + t}) d t = \int_{0}^{\infty} \frac{\partial}{\partial a} (\frac{e^{(a - 1) l n t}}{1 + t}) d t

= \int_{0}^{\infty} \frac{l n t \times t^{a - 1}}{1 + t} b u t f^{^{'}} (a) = π \times \frac{- π c o s (π a)}{{s i n}^{2} (π a)} = - π^{2} \frac{c o s (π a)}{{s i n}^{2} (π a)}

a l s o f^{(2)} (a) = \int_{0}^{\infty} \frac{{l n}^{2} (t) t^{a - 1}}{1 + t} d t a n d f^{(2)} (a) =

- π^{2} \frac{- π s i n (π a) {s i n}^{2} (π a) - 2 s i n (π a) π c o s (π a)}{{s i n}^{4} (π a)}

= π^{3} \times \frac{{s i n}^{2} (π a) + 2 c o s (π a)}{{s i n}^{3} (π a)} \Rightarrow \int_{0}^{\infty} \frac{l n (t) t^{a - 1}}{1 + t} d t = - \frac{π^{2} c o s (π a)}{{s i n}^{2} (π a)}

a n d \int_{0}^{\infty} \frac{{l n}^{2} t t^{a - 1}}{1 + t} d t = π^{3} \times \frac{{s i n}^{2} (π a) + 2 c o s (π a)}{{s i n}^{3} (π a)} .

Answered by mind is power last updated on 09/Nov/19

Commented by mathmax by abdo last updated on 10/Nov/19

t h a n k y o u s i r .

Commented by mind is power last updated on 10/Nov/19

y^{'} re welcom