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Math Help - Is my answer to this integral correct

  1. #1
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    Is my answer to this integral correct

     \int_0^{\pi} \cos(\tan(x)) ~dx ~= \frac{\pi}{e} ???
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  2. #2
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    Quote Originally Posted by simplependulum View Post

     \int_0^{\pi} \cos(\tan(x)) ~dx ~= \frac{\pi}{e} ???
    yes, it's correct. \int_0^{\pi} \cos(\tan(x)) \ dx= 2 \int_0^{\frac{\pi}{2}} \cos(\tan x) \ dx = 2\int_0^{\infty} \frac{\cos x}{1+x^2} \ dx = \frac{\pi}{e}. (a very well-known result!)
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  3. #3
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    Thank you so much Mr NonCommAlg

    This result really surprised me !
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  4. #4
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    Quote Originally Posted by NonCommAlg View Post
    yes, it's correct. \int_0^{\pi} \cos(\tan(x)) \ dx= 2 \int_0^{\frac{\pi}{2}} \cos(\tan x) \ dx = 2\int_0^{\infty} \frac{\cos x}{1+x^2} \ dx = \frac{\pi}{e}. (a very well-known result!)
    Hey! Will you please explain the last step for me? I don't understand how you converted cos(tan(x)).
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  5. #5
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    Quote Originally Posted by Kaitosan View Post
    Hey! Will you please explain the last step for me? I don't understand how you converted cos(tan(x)).
    A simple substitution x=\tan(x)
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  6. #6
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    Huh?

    cos(tan(x)) dx = cos(u) du

    du = sec^2x dx which doesn't make sense.

    How does cos(tan(x)) = cos(x)/(1+x^2)?

    Please clarify. Maybe there's some trig information I'm missing or something.
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  7. #7
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    Maybe because \sec^2(x)=\frac{1}{\cos^2(x)}=\frac{\cos^2(x)+\sin  ^2(x)}{\cos^2(x)}=1+\tan^2(x)=1+u^2



    In fact, it can be known that the derivative of tangent is also 1+\tan^2
    It depends on the way you learnt it ^^'
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  8. #8
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    Another way to look at it: u=\tan x,\ x= \arctan u,\ dx=\frac{du}{1+u^2}
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  9. #9
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    ooh I see. Thanks!

    One last question -

    How do you integrate cos(x)/(1+x^2)? Trig sub and inverse trig function integration seem not to work.
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  10. #10
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    Quote Originally Posted by Kaitosan View Post
    ooh I see. Thanks!

    One last question -

    How do you integrate cos(x)/(1+x^2)? Trig sub and inverse trig function integration seem not to work.
    Residue theorem
    That's complex analysis.
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  11. #11
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    Well. In that case, I feel better! For a moment I felt like crap.
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