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Math Help - Integration Help

  1. #1
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    Integration Help

    Hey guys, I'm new to integration and these are giving me a really rough time, even just a push in the right direction would be great. Thanks for taking a look! So far, I've only learned the basic integration techniques and u-substitution.

    1) \int\frac{x+1}{\sqrt{x^2+2x-3}}dx

    2) \int\frac{dx}{x^{\frac{3}{4}}(x^\frac{1}{4}+2)^2}

    I solved 2) and got 16\sqrt[4]{x}-\frac{4}{\sqrt[4]{x}}. Not sure if this is right, I did expanded completely and brought everything up top before integrating.

    3) \int3x\sin(4-x^2)dx

    4) \int\frac{dx}{\cos^2x\sqrt{1+\tan{x}}}

    Thanks for any pointers!
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  2. #2
    MHF Contributor arbolis's Avatar
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    For the first one, I'd try a u-substitution as u=x^2+2x-3 \Rightarrow du=2x+2 dx.
    Hence the integral is worth \frac{1}{2} \int \frac{du}{\sqrt u}. Finish it.
    Edit: I also see a u-sub for the third integral : let u=4-x^2... can you finish it?
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  3. #3
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    Quote Originally Posted by arbolis View Post
    For the first one, I'd try a u-substitution as u=x^2+2x-3 \Rightarrow du=2x+2 dx.
    Hence the integral is worth \frac{1}{2} \int \frac{du}{\sqrt u}. Finish it.
    Edit: I also see a u-sub for the third integral : let u=4-x^2... can you finish it?
    Wicked I think I worked out a solution for both of them, if you have a sec would you mind checking over my work?

    1) \int\frac{x+1}{\sqrt{x^2+2x-3}}dx

    Let u=x^2+2x-3

    du=2x+2dx

    \frac{1}{2}du=(x+1)dx

    Therefore

    \frac{1}{2}\int\frac{du}{\sqrt{u}}

    =\frac{1}{2}\int u^{-{\frac{1}{2}}}du

    =(\frac{1}{2})(2u^{\frac{1}{2}})+c

    =\sqrt{u}+c

    \boxed{=\sqrt{x^2+2x-3}+c}

    ----------------------------------------------------------
    And for number 3) I got the following;
    <br />
\int3x\sin{(4-x^2)}dx

    Let u=4-x^2

    du=-2xdx

    -\frac{3}{2}du=3xdx

    Therefore

    \int(\sin{(u)})(-\frac{3}{2}du)

    =-\frac{3}{2}\int\sin(u)du
    <br />
=-\frac{3}{2}(-\cos{u})+C
    <br />
\boxed{=\frac{3}{2}\cos{(4-x^2)}+C}

    What do you think?
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  4. #4
    MHF Contributor arbolis's Avatar
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    Both looks right to me. Very good!
    I'll let others helping you for the 2 remaining ones.
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  5. #5
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    Cool, thanks!
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