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Math Help - Rational Trignometric Integral

  1. #1
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    Rational Trignometric Integral



    hmm any help?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by usagi_killer View Post


    hmm any help?
    one of, i am sure, many ways:

    multiply by \frac {1 + \sin x}{1 + \sin x}

    you get

    \int \frac {(1 + \sin x)^2}{\cos^2 x}~dx

    Now expand the numerator and break this into 3 integrals. you should be able to handle each pretty easily
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  3. #3
    Super Member General's Avatar
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    Quote Originally Posted by usagi_killer View Post


    hmm any help?
    Multiply the integrated function by \frac{1+sin(x)}{1+sin(x)}

    With some algebra and trigonometric identities, you will get:
    \int \frac{sin^2(x)+2sin(x)+1}{cos^2(x)} dx
    =\int tan^2(x)dx + 2 \int \frac{sin(x)}{cos^2(x)} dx + \int sec^2(x) dx

    For the first: use a well-known trigometric identity.
    For the second: use a U-Substitution.
    For the third: Its a well-known integral.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by usagi_killer View Post


    hmm any help?
    \int\frac{1+\sin(x)}{1-\sin(x)}\text{ }dx=\int\frac{1+\sin(x)}{1-\sin(x)}\cdot\frac{1+\sin(x)}{1+\sin(x)}\text{ }dx= \int\frac{\left(1+\sin(x)\right)^2}{1-\sin^2(x)}\text{ }dx=\int\frac{\left(1+\sin(x)\right)^2}{\cos^2(x)}  \text{ }dx. I'm sure you can go from there.
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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by General View Post
    Multiply the integrated function by \frac{1+sin(x)}{1+sin(x)}

    With some algebra and trigonometric identities, you will get:
    \int \frac{sin^2(x)+2sin(x)+1}{cos^2(x)} dx
    =\int tan^2(x)dx + 2 \int \frac{sin(x)}{cos^2(x)} dx + \int sec^2(x) dx

    For the first: use a well-known trigometric identity.
    For the second: use a U-Substitution.
    For the third: Its a well-known integral.
    u-sub is not necessary for the second
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  6. #6
    Super Member General's Avatar
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    Quote Originally Posted by Jhevon View Post
    u-sub is not necessary for the second
    But it solves it.
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