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Math Help - How do I solve the integral x|x|dx?

  1. #1
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    How do I solve the integral x|x|dx?

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

    How should I proceed?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Ulysses View Post
    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.
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  3. #3
    Senior Member roninpro's Avatar
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    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.
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  4. #4
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    I believe

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

    You know what to do.
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