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andu1854

  • one year ago

Evaluate the following integral I will draw it out, but its set up like this: sin^2 x dx ---------- sqrt(1-cosx) I need help setting up what u will equal and going from there...

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  1. andu1854
    • one year ago
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    |dw:1434089857664:dw|

  2. andu1854
    • one year ago
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    From here, I was able to conjugate the denominator to get: |dw:1434089965873:dw|

  3. ganeshie8
    • one year ago
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    i think below identity might simplify the integrand \[1-\cos x ~=~ 2\sin^2(\frac{x}{2})\]

  4. ganeshie8
    • one year ago
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    also this \[\sin x ~~=~~2\sin(\frac{x}{2})\cos(\frac{x}{2}) \]

  5. ganeshie8
    • one year ago
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    play wid them and see if you can get a nice looking expression for the integrand

  6. anonymous
    • one year ago
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    @andu1854 I think you meant to multiply by the conjugate \(\sqrt{1+\cos x}\)? So you have \[\int_{\pi/3}^{\pi/2}\frac{\sin^2x}{\sqrt{1-\cos x}}\,dx=\int_{\pi/3}^{\pi/2}\frac{\sin^2x\sqrt{1+\cos x}}{\sqrt{1-\cos^2 x}}\,dx\] The denominator simplifies nicely: \[\sqrt{1-\cos^2x}=\sqrt{\sin^2x}=|\sin x|=\sin x\] where the last equality holds because \(\sin x>0\) for \(\dfrac{\pi}{3}\le x\le\dfrac{\pi}{2}\). The integral simplifies further: \[\int_{\pi/3}^{\pi/2}\frac{\sin^2x\sqrt{1+\cos x}}{\sin x}\,dx=\int_{\pi/3}^{\pi/2}\sin x\sqrt{1+\cos x}\,dx\] and you can deal with this using a simple substitution.

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