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Math Help - very hard integration question

  1. #1
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    very hard integration question

    I need some help here, this is a hard one. Show me some steps.

    For the following integral
     I = \int^{10}_{0}\frac{\sqrt{x}}{\sqrt{x}+{\sqrt{10-x}}}
    use a substitution to show that
     I = \int^{10}_{0}\frac{\sqrt{10-x}}{\sqrt{x}+{\sqrt{10-x}}}
    Use these two representations of I to evaluate I.
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  2. #2
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    Quote Originally Posted by gammaman View Post
    I need some help here, this is a hard one. Show me some steps.

    For the following integral
     I = \int^{10}_{0}\frac{\sqrt{x}}{\sqrt{x}+{\sqrt{10-x}}} \, {\color{red}dx}
    use a substitution to show that
     I = \int^{10}_{0}\frac{\sqrt{10-x}}{\sqrt{x}+{\sqrt{10-x}}} \, {\color{red}dx}
    Use these two representations of I to evaluate I.
    First note what I've added in red. Sloppy notation (such as your omissions) always leads to difficulty and trouble and lost marks.


    Substitute x = 10 - u to get  I = \int^{10}_{0}\frac{\sqrt{10-u}}{\sqrt{u}+{\sqrt{10-u}}} \, du.

    Note that u is just a dummy variable in the definite integral and can be therefore replaced with any symbol, including x ....

    Now add the two expressions for I to get 2I = \int_0^{10} dx \, ....
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