Math, asked by vivek8375, 6 months ago

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

Answers

Answered by Anonymous
144

Answer:

\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> }

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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\: &amp;\: dx =xdt}

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

 = log |t| + c

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

 = log |1 + logx| + ctt

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Answered by Anonymous
1

According to the question

⟹ \frac{1}{x + xlogx} = \frac{1}{x(1 + logx)} \\  \\ </p><p></p><p>\bold{\red{Let 1+logx=t}} \\  \\ </p><p></p><p>\bold{Differentiating\: both \:sides \:w.r.t.x} \\  \\ [/t ex] [tex]⟹ 0 + \frac{1}{x} = \frac{dt}{dx} </p><p></p><p>⟹ \frac{1}{x} = \frac{dt}{dx} \\  \\ </p><p></p><p></p><p></p><p>\bold{Integrating function:-} \\  \\ </p><p></p><p>⟹∫ \frac{1}{x + xlogx} dx = ∫ \frac{1}{x(1 + logx)}  \\  \\ </p><p></p><p>\bold{Putting \:1+logx\: &amp;\: dx =xdt} \\  \\ </p><p></p><p>= ∫ \frac{1}{x(t)} dt \times x = ∫ \frac{1}{t}  \\  \\ </p><p></p><p>= log |t| + c=log∣t∣+c \\  \\ </p><p></p><p>\bold{Put\: t=1+logx} \\  \\ </p><p></p><p>= log |1 + logx| + c=log∣1+logx∣+c tt</p><p></p><p>

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