Integrate the function: ( 1 + x ) ^2 x - Vidyadip

Question

Integrate the function: $\frac{{{{\left( {1 + \log x} \right)}^2}}}{x}$

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Answer

Let $I = \int {\frac{{{{\left( {1 + \log x} \right)}^2}}}{x}dx}$ ...(i)

Putting $1 + \log x = t$

$ \Rightarrow \frac{1}{x} = \frac{{dt}}{{dx}}$

$\Rightarrow \frac{{dx}}{x} = dt$

$\therefore$ From eq. (i), $I = \int {{t^2}dt} $

$= \frac{{{t^3}}}{3} + c$

$= \frac{1}{3}{\left( {1 + \log x} \right)^3} + c$

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