1-explicite-f-a-0-t-a-1-lnt-1-t-dt-with-0-lt-a-lt-1-2-calculate-0-lnt-1-t-t-dt-

Question Number 121862 by Bird last updated on 12/Nov/20

1) e x p l i c i t e f (a) = \int_{0}^{\infty} \frac{t^{a - 1} l n t}{1 + t} d t

w i t h 0 < a < 1

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

Answered by mnjuly1970 last updated on 12/Nov/20

s o l u t i o n : 1 :: g (b) = \int_{0}^{\infty} \frac{t^{a + b - 1}}{1 + t} d t

f (a) = g^{'} (0)

g (b) = Γ (a + b) Γ (1 - a - b) = \frac{π}{s i n (π (a + b))}

g^{'} (b) = \frac{- π^{2} c o s (π (a + b))}{{s i n}^{2} (π (a + b))}

Missing \left or extra \right

f (a) = - π^{2} c o t (π a) c s c (π a) \dots

2 :: f (\frac{1}{2}) = 0

w e k n o w t h a t ::

\int_{0}^{\infty} \frac{l n (x)}{1 + x^{2}} d x \overset{e a s y}{=} 0