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Thread: U-subsitution

  1. #1
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    Question U-subsitution

    Can someone please explain the u-substitution?

    I don't see a tan(x/2) anywhere in the integral expression.

    U-subsitution-screen-shot-2019-07-21-10.08.42-pm.png
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  2. #2
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    Re: U-subsitution

    well looking at the solution the substitution will get us the form

    $I(x)= \dfrac{\dfrac{d}{dx} \tan\left(\dfrac x 2\right)}{\tan\left(\dfrac x 2\right)}$ for the integrand

    $I(x) = \dfrac{\frac{1}{2} \sec ^2\left(\dfrac{x}{2}\right)}{\tan\left(\dfrac x 2\right)} =$

    $\dfrac{1}{2\cos^2\left(\dfrac x 2\right)\tan\left(\dfrac x 2\right)}=$

    $\dfrac{1}{2\sin\left(\dfrac x 2\right)\cos\left(\dfrac x 2\right)} = $

    $\dfrac{1}{\sin(x)}$

    which is the original integrand. So the substitution is correct.
    Thanks from topsquark
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