How do I solve the integral x|x|dx?

**Ulysses** · July 3rd 2010, 12:48 PM

I must solve $\displaystyle\int_{}^{}x|x|dx$

How should I proceed?

**Jhevon** · July 3rd 2010, 01:05 PM

Originally Posted by Ulysses

I must solve $\displaystyle\int_{}^{}x|x|dx$

How should I proceed?

hmm, it would depend on the limits. $\displaystyle \int x |x|~dx = \left \{ \begin{array}{lr} \int x^2 ~dx & \text{ if } x \ge 0 \\ & \\ - \int x^2~dx & \text{ if } x < 0 \end{array} \right.$

**roninpro** · July 3rd 2010, 01:11 PM

You can write $|x|=\sqrt{x^2}$ . Then, you have $\int x|x|\text{ d}x=\int x\sqrt{x^2}\text{ d}x$ . Let $u=x^2$ and $\text{d}u=2x\text{ d}x$ . Your integral is $\frac{1}{2}\int \sqrt{u}\text{ d}u=\frac{1}{3}u^{3/2}+C=\frac{1}{3}(x^2)^{3/2}+C=\frac{1}{3}|x|^3+C$ .

**theodds** · July 3rd 2010, 01:11 PM

I believe

$\displaystyle<br /> \frac{d}{dx} |x|^3 = 3x|x|.<br />$

You know what to do.

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