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Math Help - How to solve Integrals?

  1. #1
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    How to solve Integrals?

    Hi! Can someone please walk me through on how to answer these integrals?

    1. How to solve Integrals?-integral-1.png
    2. How to solve Integrals?-integral-2.png
    3. How to solve Integrals?-integral-3.png

    Thanks! :-)
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  2. #2
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    Re: How to solve Integrals?

    1. Make the substitution t = (u-3)^2 - 8

    2. Try integration by parts, so that inside the integral sign, you differentiate ln(x) and integrate sqrt(x), the product of these two will be a power function.

    3. Make the substitution s = u-3, multiply out the square, and then you will have a sum of power functions.
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  3. #3
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    Re: How to solve Integrals?

    Quote Originally Posted by Hi888 View Post
    Hi! Can someone please walk me through on how to answer these integrals?

    1. Click image for larger version. 

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    2. Click image for larger version. 

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    3. Click image for larger version. 

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    Thanks! :-)
    Here's an easier substitution in the first...

    \displaystyle \begin{align*} \int{\frac{1}{3 + \sqrt{t + 8}}\,dt} &= \int{\frac{2\sqrt{t + 8}}{2\sqrt{t + 8}\left( 3 + \sqrt{t+ 8}\right)} \,dt} \\ &= \int{\frac{2u}{3 + u}\,du} \textrm{ after making the substitution }u = \sqrt{t + 8} \implies du = \frac{1}{2\sqrt{t + 8}}\,dt \\ &= \int{\frac{2(v - 3)}{v}\,dv} \textrm{ after making the substitution } v = 3 + u \implies dv = du \\ &= \int{\frac{2v - 6}{v}\,dv} \\ &= \int{2 - \frac{6}{v}\,dv}  \end{align*}

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