Question

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

Answer 1

Step-by-step explanation:

$\huge\mathfrak\green{\bold{\underline{☘{ ℘ɧεŋσɱεŋศɭ}☘}}}$

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

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

$\huge\tt\underline\pink{꧁Answer꧂</p><p> }$

╔════════════════════════╗

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _✍️

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

Let 1+logx=t

Differentiating both sides w.r.t.x

$⟹</p><p>0 + \frac{1}{x} = \frac{dt}{dx}$

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

$dx = xdt$

Integrating function:-

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

Putting 1+logx & dx =xdt

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

$= log |t| + c$

Put t=1+logx

$= log |1 + logx| + c$

╚════════════════════════╝

нσρє ıт нєłρs yσυ

_____________________

тнαηkyσυ

Answer 2

Step-by-step explanation:

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

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

$\sf= log |t| + c$

$\sf\red{= log |1 + logx| + c}$