help-me-please-ln-x-1-x-dx-

Question Number 146035 by KONE last updated on 10/Jul/21

h e l p m e p l e a s e

\int \frac{l n (x + 1)}{x} d x = ? ?

Answered by KONE last updated on 10/Jul/21

p l e a s e

Answered by puissant last updated on 10/Jul/21

\frac{1}{1 + x} = \underset{n = 0}{\sum^{\infty}} {(- 1)}^{n} x^{n}

\Rightarrow \ln (1 + x) = \int \underset{n = 0}{\sum^{\infty}} {(- 1)}^{n} x^{n} dx

\Rightarrow \ln (1 + x) = \underset{n = 0}{\sum^{\infty}} {(- 1)}^{n} \frac{x^{n + 1}}{n + 1} + c

x = 0 \Rightarrow \ln (1 + x) = \underset{n = 0}{\sum^{\infty}} {(- 1)}^{n} \frac{x^{n + 1}}{n + 1}

\Rightarrow \frac{\ln (1 + x)}{x} = \underset{n = 0}{\sum^{\infty}} {(- 1)}^{n} \frac{x^{n}}{n + 1}

\Rightarrow \int \frac{\ln (1 + x)}{x} dx = \underset{n = 0}{\sum^{\infty}} {(- 1)}^{n} \frac{1}{n + 1} \int x^{n} dx

\Rightarrow I = \underset{n = 0}{\sum^{\infty}} {(- 1)}^{n} \frac{x^{n + 1}}{{(n + 1)}^{2}} + k . .

Commented by KONE last updated on 12/Jul/21

t h a n k s \dots

s v p e s t p o s s i b l e d e f a i r e s a n s u t i l i s e r l e D L ?

Facebook Tweet Pin