`int (ln(x))^2/x dx` Find the indefinite integral.

Sunday, November 21, 2010

$\displaystyle\int\frac{{{\left({\ln{{\left({x}\right)}}}\right)}}^{{2}}}{{x}}{\left.{d}{x}\right.}$ Find the indefinite integral.

$\displaystyle\int\frac{{{\left({\ln{{\left({x}\right)}}}\right)}}^{{2}}}{{x}}{\left.{d}{x}\right.}$

To solve, apply u-substitution method.

Let,

$\displaystyle{u}={\ln{{x}}}$

Then, differentiate it.

$\displaystyle{d}{u}=\frac{{1}}{{x}}{\left.{d}{x}\right.}$

Plug-in them to the integral.

$\displaystyle\int\frac{{{\left({\ln{{\left({x}\right)}}}\right)}}^{{2}}}{{x}}{\left.{d}{x}\right.}$

$\displaystyle=\int{{\left({\ln{{\left({x}\right)}}}\right)}}^{{2}}\cdot\frac{{1}}{{x}}{\left.{d}{x}\right.}$

$\displaystyle=\int{{u}}^{{2}}{d}{u}$

Then, apply the formula, $\displaystyle\int{{x}}^{{n}}{\left.{d}{x}\right.}=\frac{{{x}}^{{{n}+{1}}}}{{{n}+{1}}}+{C}$ .

$\displaystyle=\frac{{{u}}^{{3}}}{{3}}+{C}$

And, substitute back $\displaystyle{u}={\ln{{x}}}$ .

$\displaystyle=\frac{{{\left({\ln{{\left({x}\right)}}}\right)}}^{{3}}}{{3}}+{C}$

Therefore, $\displaystyle\int\frac{{{\left({\ln{{\left({x}\right)}}}\right)}}^{{2}}}{{x}}{\left.{d}{x}\right.}=\frac{{{\left({\ln{{\left({x}\right)}}}\right)}}^{{3}}}{{3}}+{C}$ .

Ninenotes

Sunday, November 21, 2010

$\displaystyle\int\frac{{{\left({\ln{{\left({x}\right)}}}\right)}}^{{2}}}{{x}}{\left.{d}{x}\right.}$ Find the indefinite integral.

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