Question

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

Answer 1

Step-by-step explanation:

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

$= log |1 + logx| + c$

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Answer 2

Answer:

Step-by-step explanation

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

☘℘ɧεŋσɱεŋศɭ☘

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

❥Question᎓

integrate the function :

\frac{1}{x + xlogx}

x+xlogx

1

\huge\tt\underline\pink{꧁Answer꧂ }

꧁Answer꧂

╔════════════════════════╗

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _✍️

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

x+xlogx

1

=

x(1+logx)

1

Let 1+logx=t

Differentiating both sides w.r.t.x

⟹ 0 + \frac{1}{x} = \frac{dt}{dx}⟹0+

x

1

=

dx

dt

⟹ \frac{1}{x} = \frac{dt}{dx}⟹

x

1

=

dx

dt

dx = xdtdx=xdt

Integrating function:-

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

x+xlogx

1

dx=∫

x(1+logx)

1

dx

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

Put t=1+logx

= log |1 + logx| + c=log∣1+logx∣+c

╚════════════════════════╝

нσρє ıт нєłρs yσυ

_____________________

тнαηkyσυ