Blog: `int (ln x)^2 dx` Evaluate the integral

Saturday, June 15, 2013

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

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

To evaluate, apply integration by parts $\displaystyle\int{u}{d}{v}={u}{v}-{v}{d}{u}$ .

So let

$\displaystyle{u}={{\left({\ln{{x}}}\right)}}^{{2}}$

and

$\displaystyle{d}{v}={\left.{d}{x}\right.}$

Then, differentiate u and integrate dv.

$\displaystyle{d}{u}={2}{\ln{{x}}}\cdot\frac{{1}}{{x}}{\left.{d}{x}\right.}=\frac{{{2}{\ln{{x}}}}}{{x}}{\left.{d}{x}\right.}$

and

$\displaystyle{v}=\int{\left.{d}{x}\right.}={x}$

Plug-in them to the formula. So the integral becomes:

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

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

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

To take integral of ln x, apply integration by parts again.

So let

$\displaystyle{u}_{{2}}={\ln{{x}}}$

and

$\displaystyle{d}{v}_{{2}}={\left.{d}{x}\right.}$

Then, differentiate u and integrate dv.

$\displaystyle{d}{u}_{{2}}=\frac{{1}}{{x}}{\left.{d}{x}\right.}$

and

$\displaystyle{v}_{{2}}=\int{\left.{d}{x}\right.}={x}$

So the integral becomes:

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

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

$\displaystyle={x}{{\left({\ln{{x}}}\right)}}^{{2}}-{2}{\left({x}{\ln{{x}}}-{x}\right)}$

$\displaystyle={x}{{\left({\ln{{x}}}\right)}}^{{2}}-{2}{x}{\ln{{x}}}+{2}{x}$

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

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Saturday, June 15, 2013

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

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