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Math Help - I'm stuck, what should I do next?

  1. #1
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    I'm stuck, what should I do next?

    \int\frac{tan\frac{x}{2}}{1+tan^{2}\frac{x}{2}}
    =2\int\frac{tan\frac{x}{2}}{1+tan^{2}\frac{x}{2}}

    what should I do next?
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  2. #2
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    Quote Originally Posted by honestliar View Post
    \int\frac{tan\frac{x}{2}}{1+tan^{2}\frac{x}{2}}
    =2\int\frac{tan\frac{x}{2}}{1+tan^{2}\frac{x}{2}}

    what should I do next?
    First, what did you do? The two are obviously NOT equal. The second is twice the first.
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    First, what did you do? The two are obviously NOT equal. The second is twice the first.
    I used u-substitution for the numerator.

    should I change the denominator to sec^{2}\frac{x}{2}?
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  4. #4
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    Quote Originally Posted by honestliar View Post
    I used u-substitution for the numerator.

    should I change the denominator to sec^{2}\frac{x}{2}?
    Ah, well if you're using u-substitution it's especially important to specify the unit of integration at the end of your expression. The above confusion is evidence of what can go wrong when this isn't specified.

    And off the top of my head, I think changing it to sec^2 would be a good idea, it might allow you to express the whole fraction as some form of sin^2 on cos^2 or vice versa.
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  5. #5
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    Quote Originally Posted by honestliar View Post
    I used u-substitution for the numerator.

    should I change the denominator to sec^{2}\frac{x}{2}?
    Yes. From there you can simplify your integral:

    \int\frac{tan(\frac{x}{2})}{sec^2(\frac{x}{2})}dx\  Rightarrow

    \int\frac{sin(\frac{x}{2})cos^{2}(\frac{x}{2})}{co  s(\frac{x}{2})}dx\Rightarrow

    \int\sin(\frac{x}{2})cos(\frac{x}{2})dx

    From here, you can use u-du substitution. Do you see which one you should take?

    From your earlier work I'm assuming you did:
    u=tan(\frac{x}{2})

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

    2du=sec^{2}(\frac{x}{2})dx

    Which is cool, except you don't have a sec^{2}(\frac{x}{2}) in your integral, you have a \frac{1}{sec^{2}(\frac{x}{2})}
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