Integral that I just can't figure out !

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March 8th 2013, 03:00 PM
uberstu

Integral that I just can't figure out !

Hi all, for some reason I just keep going round in circles with this indefinite integral;

Int: (2x^3 + x) * arctan[x] dx

Sorry for lack of formatting. This is driving me mad, I know it should be jumping out at me what to do.

Thanks in advance for your help !!!

Stu
March 8th 2013, 03:08 PM
Prove It

Re: Integral that I just can't figure out !

$\displaystyle \begin{align*} \int{\left( 2x^3 + x \right) \arctan{(x)}\,dx} &= \left( \frac{x^4}{2} + \frac{x^2}{2} \right) \arctan{(x)} - \int{ \left( \frac{x^4}{2} + \frac{x^2}{2} \right)\left( \frac{1}{1 + x^2} \right) dx } \\ &= \frac{x^2}{2}\left( x^2 + 1 \right) \arctan{(x)} - \frac{1}{2} \int{ \frac{x^2 \left( x^2 + 1 \right)}{1 + x^2}\, dx } \\ &= \frac{x^2}{2} \left( x^2 + 1 \right) \arctan{(x)} - \frac{1}{2} \int{ x^2\,dx } \\ &= \frac{x^2}{2} \left( x^2 + 1 \right) \arctan{(x)} - \frac{x^3}{6} + C \end{align*}$
March 8th 2013, 04:07 PM
uberstu

Re: Integral that I just can't figure out !

Thankyou :)

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