logx-1-x-x-dx-please-help-this-

Question Number 49003 by mondodotto@gmail.com last updated on 01/Dec/18

\int \frac{\log x}{\sqrt{1 - x^{x}}} d x

please help this

Commented by maxmathsup by imad last updated on 01/Dec/18

I = \int \frac{l n (x)}{\sqrt{1 - e^{x l n (x)}}} d x c h a n g e m e n t x l n (x) = t g i v e (l n (x) + 1) d x = d t

I = \int \frac{l n (x) + 1 - 1}{\sqrt{1 - x^{x}}} d x = \int \frac{l n (x) + 1}{\sqrt{1 - e^{x l n x}}} d x - \int \frac{d x}{\sqrt{1 - x^{x}}}

= \int \frac{d t}{\sqrt{1 - t}} - \int \frac{d x}{\sqrt{1 - x^{x}}} = - 2 \sqrt{1 - t} - \int \frac{d x}{\sqrt{1 - x^{x}}} = - 2 \sqrt{1 - x^{x}} - \int \frac{d x}{\sqrt{1 - x^{x}}}

l e t f i n d \int \frac{d x}{\sqrt{1 - x^{x}}} a t f o r m o f s e r i e d u e t o 0 < x^{x} < 1

w e h a v e {(1 + u)}^{α} = 1 + \frac{α u}{1!} + \frac{α (α - 1)}{2!} u^{2} + \dots . + \frac{α (α - 1) \dots (α - n + 1)}{n!} u^{n} + \dots \Rightarrow

{(1 - u)}^{α} = 1 - α u + \frac{α (α - 1)}{2!} u^{2} + \frac{α (α - 1) \dots (α - n + 1)}{n!} {(- 1)}^{n} u^{n} + \dots \Rightarrow

{(1 - x^{x})}^{- \frac{1}{2}} = 1 + \frac{1}{2} x^{x} + \frac{(- \frac{1}{2}) (- \frac{3}{2})}{2!} x^{2 x} + \dots \frac{(- \frac{1}{2}) (- \frac{3}{2}) \dots (- \frac{1}{2} - n + 1)}{n!} {(- 1)}^{n} x^{n x} + \dots

\Rightarrow \int \frac{d x}{\sqrt{1 - x^{x}}} = x + \frac{1}{2} \int x^{x} d x + \int \frac{3}{4 2!} x^{2 x} d x + \dots

+ \frac{(- \frac{1}{2}) (- \frac{3}{2}) \dots . (- \frac{1}{2} - n + 1)}{n!} {(- 1)}^{n} \int x^{n x} d x + \dots .

Commented by MJS last updated on 01/Dec/18

where did you find this ?

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