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Math Help - Transform these integrands

  1. #1
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    Transform these integrands

    Hi,

    I am having trouble at transforming integrands. I just need these to be transformed then I think I will be able to integrate it. I hope you can help me.



    Thank you.
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  2. #2
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    Re: Transform these integrands

    Quote Originally Posted by reiward View Post
    Hi,

    I am having trouble at transforming integrands. I just need these to be transformed then I think I will be able to integrate it. I hope you can help me.



    Thank you.
    For the first, long divide.

    For the second

    \displaystyle \begin{align*} \int{\frac{2t - 3}{\sqrt{4t - 1}}\,dt} &= \frac{1}{4}\int{\frac{2t-3}{\sqrt{4t-1}}\,4\,dt} \end{align*}

    Make the substitution \displaystyle \begin{align*} u = 4t - 1 \implies du = 4\,dt \end{align*} and the integral becomes

    \displaystyle \begin{align*} \frac{1}{4}\int{\frac{2t-3}{\sqrt{4t-1}}\,4\,dt} &= \frac{1}{4}\int{\frac{\frac{1}{2}\left(u + 1\right) - 3}{\sqrt{u}}\,du} \\ &= \frac{1}{8}\int{\frac{u + 1 - 6}{\sqrt{u}}\,du} \\ &= \frac{1}{8}\int{\frac{u - 5}{\sqrt{u}}\,du} \\ &= \frac{1}{8}\int{u^{\frac{1}{2}} - 5u^{-\frac{1}{2}}\,du} \end{align*}

    Go from here.
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