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Math Help - Integration with substitution

  1. #1
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    Integration with substitution

    Hey I was wondering if I can have some help with an integration problem. The integral is cos(x)/sqrt(x) dx. They tell me to use the substitute u = 2 sqrt(x).

    The u is nice because when we find du = 1/sqrt(x) dx. Now im stuck with the integral of cos(x) du.

    I tried to solve for x in terms of u and x = (u^2)/4

    Replacing I get the integral of cos(u^2/4)du.

    This is where im stuck.

    Any help is appreciated. Thanks
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  2. #2
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    Quote Originally Posted by jlip View Post
    Hey I was wondering if I can have some help with an integration problem. The integral is cos(x)/sqrt(x) dx. They tell me to use the substitute u = 2 sqrt(x).

    The u is nice because when we find du = 1/sqrt(x) dx. Now im stuck with the integral of cos(x) du.

    I tried to solve for x in terms of u and x = (u^2)/4

    Replacing I get the integral of cos(u^2/4)du.

    This is where im stuck.

    Any help is appreciated. Thanks
    Are you sure it's not cos(x) sqrt(x) dx ....?
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  3. #3
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    yep it is cos(x) divided by the square root of x
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  4. #4
    Super Member Matt Westwood's Avatar
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    You can use the identity that converts \cos^2 x into something involving \cos 2x - it should be at your fingertips but unfortunately it's not at mine!

    Then you'll have something in the form \cos 2u or something, and you should be able to integrate that.
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  5. #5
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    Quote Originally Posted by Matt Westwood View Post
    You can use the identity that converts \cos^2 x into something involving \cos 2x - it should be at your fingertips but unfortunately it's not at mine!

    Then you'll have something in the form \cos 2u or something, and you should be able to integrate that.
    No. If the problem is correct (and I have my doubts that it is), then you have to integrate \cos (u^2). This is not the same as \cos^2 (u). In fact, \int \cos (u^2) \, du cannot be found using a finite sum of elementary functions (which is what I'm assuming the OP wants to do). It's a Fresnel integral: Fresnel integral - Wikipedia, the free encyclopedia
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  6. #6
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by mr fantastic View Post
    No. If the problem is correct (and I have my doubts that it is), then you have to integrate \cos (u^2). This is not the same as \cos^2 (u). In fact, \int \cos (u^2) \, du cannot be found using a finite sum of elementary functions (which is what I'm assuming the OP wants to do). It's a Fresnel integral: Fresnel integral - Wikipedia, the free encyclopedia
    D'oh! Sorry, goes to show how useful LaTeX is in making things clearer. Reading raw unrendered code is trickier than it looks.
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