Question

integrate the function :
$\frac{1}{x + xlogx}$
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Answer 1

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$\red{\bold{\underline{\underline{❥Question᎓}}}}$integrate the function :

$\frac{1}{x + xlogx}$

$\huge\huge\tt\blue{「Answer」</p><p> }$

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$⟹ \frac{1}{x + xlogx} = \frac{1}{x(1 + logx)} </p><p>$

$\bold{\red{Let 1+logx=t}}$

$\bold{Differentiating\: both \:sides \:w.r.t.x}[/t ex]</p><p>[tex]⟹</p><p>0 + \frac{1}{x} = \frac{dt}{dx}$

$⟹</p><p> \frac{1}{x} = \frac{dt}{dx}$

$dx = xdt$

$\bold{Integrating function:-}$

$⟹∫ \frac{1}{x + xlogx} dx = ∫ \frac{1}{x(1 + logx)} dx</p><p>$

$\bold{Putting \:1+logx\: &\: dx =xdt}$

$= ∫ \frac{1}{x(t)} dt \times x = ∫ \frac{1}{t} dt$

$= log |t| + c$

$\bold{Put\: t=1+logx}$

$= log |1 + logx| + c$

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Answer 2

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$\huge\pink{\boxed{\green{\mathfrak{\fcolorbox{red}{aqua}{\underline{\pink{Answer}}}}}}}$

log|1+logx|+c

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please refer to the attachment.

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hope it helps.