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Math Help - Integrating sin(sqrt x)?

  1. #1
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    Integrating sin(sqrt x)?

    I need to solve the problem:

    \int\sin\sqrt x\ dx

    Using integration by parts. So far, I've done:

    u=\sqrt x
    du=\frac{1}{2\sqrt x}\ dx
    2udu=dx

    But then I'm lost.
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  2. #2
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    Hello, quarks!

    You're off to a good start . . .


    \int\sin(\sqrt{x})\,dx

    So far, I've done:

    w\:=\:\sqrt x \quad\Rightarrow x \:=\:w^2\quad\Rightarrow\quad dx \:=\:2w\,dw

    Substitute: . \int 2w\sin w\,dw

    Integrate by parts:
    . . \begin{array}{ccccccc}u &=&2w & &dv &=&\sin w\,dw \\ du &=&2\,dw & & v &=&-\cos w\end{array}

    And we have: . -2w\cos w + 2\int\cos w\,dw \;=\;-2w\cos w + 2\sin w + C

    Back-substitute: . -2\sqrt{x}\cos(\sqrt{x}) + 2\sin(\sqrt{x}) + C

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