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Math Help - Can you integrate this?

  1. #1
    Senior Member vincisonfire's Avatar
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    Can you integrate this?

    I can't find how to integrate this
    x^2*sqrt(2x-x^2)
    I tried with part derivation but I can't make it.
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  2. #2
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    Hello, vincisonfire!

    \int x^2\sqrt{2x-x^2}\,dx

    We have: . 2x - x^2 \;=\;{\color{blue}1 - 1} + 2x - x^2 \;=\;1 - (x^2-2x+1) \;=\;1-(x-1)^2

    The integral becomes: . \int x^2\sqrt{1-(x-1)^2}\,dx

    Let: x-1 \:=\:\sin\theta \quad\Rightarrow\quad x \:=\:\sin\theta + 1 \quad\Rightarrow\quad dx \:=\:\cos\theta\,d\theta

    . . and: . \sqrt{1-(x-1)^2} \;=\;\sqrt{1-\sin^2\!\theta} \;=\;\sqrt{\cos^2\!\theta} \;=\;\cos\theta

    Substitute: . \int(\sin\theta+1)^2\cdot\cos\theta\cdot\cos\theta  \,d\theta \;=\;\int\left(\sin^2\!\theta + 2\sin\theta + 1\right)\cos^2\!\theta\,d\theta

    . . and we have: . \int\bigg(\sin^2\!\theta\cos^2\!\theta + 2\sin\theta\cos^2\!\theta + \cos^2\!\theta\bigg)\,d\theta


    And I assume you can finish it now . . .

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