Question-70909

Question Number 70909 by rajesh4661kumar@gmail.com last updated on 09/Oct/19

Commented by rajesh4661kumar@gmail.com last updated on 10/Oct/19

28

Commented by Abdo msup. last updated on 10/Oct/19

f (t) = \int_{0}^{1} \frac{x^{t} - 1}{l n x} d x c h a n g e m e n t l n x = - u g i v e

f (t) = - \int_{0}^{\infty} \frac{e^{- t u} - 1}{- u} (- e^{- u}) d u

= - \int_{0}^{\infty} \frac{e^{- (t + 1) u} - e^{- u}}{u} d u = \int_{0}^{\infty} \frac{e^{- u} - e^{- (t + 1) u}}{u} d u

f^{^{'}} (t) = \int_{0}^{\infty} e^{- (t + 1) u} d u = {[- \frac{1}{t + 1} e^{- (t + 1) u}]}_{u = 0}^{\infty}

= \frac{1}{t + 1} \Rightarrow f (t) = l n (t + 1) + c

f (o) = 0 = c \Rightarrow f (t) = l n (t + 1) (t ⩾ 0)

Answered by Henri Boucatchou last updated on 10/Oct/19

W h a t q u e s t i o n (s) ?