∫(1/(1+ln x))=?


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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
	$t=1+lnx x=e^(t−1) dx=e^(t−1) dt ∫((e^(t−1) dt)/t) (1/e)∫((1+t+(t^2 /(2!))+(t^3 /(3!))+(t^4 /(4!))+...)/t)dt (1/e)∫(1/t)+(1/1)+(t/(2!))+(t^2 /(3!))+(t^3 /(4!))+... dt (1/e)(lnt+t+(t^2 /(2×2!))+(t^3 /(3×3!))+(t^4 /(4×4!))+...)+c (1/e){ln(1+lnx)+(1+lnx)+(((1+lnx)^2 )/(2×2!))+(((1+lnx)^3 )/(3×3!))+..}+c$
	$t = 1 + l n x x = e^{t - 1} d x = e^{t - 1} d t$ $\int \frac{e^{t - 1} d t}{t}$ $\frac{1}{e} \int \frac{1 + t + \frac{t^{2}}{2!} + \frac{t^{3}}{3!} + \frac{t^{4}}{4!} + . . .}{t} d t$ $\frac{1}{e} \int \frac{1}{t} + \frac{1}{1} + \frac{t}{2!} + \frac{t^{2}}{3!} + \frac{t^{3}}{4!} + . . . d t$ $\frac{1}{e} (l n t + t + \frac{t^{2}}{2 \times 2!} + \frac{t^{3}}{3 \times 3!} + \frac{t^{4}}{4 \times 4!} + . . .) + c$ $\frac{1}{e} {l n (1 + l n x) + (1 + l n x) + \frac{{(1 + l n x)}^{2}}{2 \times 2!} + \frac{{(1 + l n x)}^{3}}{3 \times 3!} + . .} + c$
	Commented by arvinddayama01@gmail.com last updated on 07/Oct/18

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

	$thanx sir$
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