1-1-ln-x-

Question Number 44921 by manish09@gmail.com last updated on 06/Oct/18

\int \frac{1}{1 + \ln x} = ?

Commented by MJS last updated on 06/Oct/18

see question 44674

Answered by tanmay.chaudhury50@gmail.com last updated on 06/Oct/18

t = 1 + l n x x = e^{t - 1} d x = e^{t - 1} d t

\int \frac{e^{t - 1} d t}{t}

\frac{1}{e} \int \frac{1 + t + \frac{t^{2}}{2!} + \frac{t^{3}}{3!} + \frac{t^{4}}{4!} + \dots}{t} d t

\frac{1}{e} \int \frac{1}{t} + \frac{1}{1} + \frac{t}{2!} + \frac{t^{2}}{3!} + \frac{t^{3}}{4!} + \dots d t

\frac{1}{e} (l n t + t + \frac{t^{2}}{2 \times 2!} + \frac{t^{3}}{3 \times 3!} + \frac{t^{4}}{4 \times 4!} + \dots) + c

\frac{1}{e} {l n (1 + l n x) + (1 + l n x) + \frac{{(1 + l n x)}^{2}}{2 \times 2!} + \frac{{(1 + l n x)}^{3}}{3 \times 3!} + . .} + c

Commented by arvinddayama01@gmail.com last updated on 07/Oct/18

Thanks

Commented by manish09@gmail.com last updated on 07/Oct/18

thanx sir