Question

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

Answer 1

Step-by-step explanation:

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

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

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

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

╔════════════════════════╗

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _✍️

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

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

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

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

$dx = xdt$

$\bold{Integrating function:-}$

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

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

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

$= log |t| + c$

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

$= log |1 + logx| + c$

╚════════════════════════╝

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Answer 2

Step-by-step explanation:

Step-by-step explanation:

\huge{\bold☘}\mathfrak\pink{\bold{\underline{{ ℘ɧεŋσɱεŋศɭ}}}}{\bold☘}☘

℘ɧεŋσɱεŋศɭ

☘

\red{\bold{\underline{\underline{❥Question᎓}}}}

❥Question᎓

integrate the function :

\frac{1}{x + xlogx}

x+xlogx

1

\huge\huge\tt\blue{「Answer」 }「Answer」

╔════════════════════════╗

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _✍️

⟹ \frac{1}{x + xlogx} = \frac{1}{x(1 + logx)}⟹

x+xlogx

1

=

x(1+logx)

1

\bold{\red{Let 1+logx=t}}Let1+logx=t

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

x

1

=

dx

dt

⟹ \frac{1}{x} = \frac{dt}{dx}⟹

x

1

=

dx

dt

dx = xdtdx=xdt

\bold{Integrating function:-}Integratingfunction:−

⟹∫ \frac{1}{x + xlogx} dx = ∫ \frac{1}{x(1 + logx)} dx⟹∫

x+xlogx

1

dx=∫

x(1+logx)

1

dx

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

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

x(t)

1

dt×x=∫

t

1

dt

= log |t| + c=log∣t∣+c

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

= log |1 + logx| + c=log∣1+logx∣+c

╚════════════════════════╝

нσρє ıт нєłρs yσυ

_____________________

тнαηkyσυ