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Math Help - integral (square root)

  1. #1
    Member mybrohshi5's Avatar
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    Unhappy integral (square root)

    I am having trouble solving this integral
     \int ^2_1 \sqrt(4x^2+4+\frac{1}{x^2})  dx

    I think i can reduce the equation under the square root to an expression that is squared which would get rid of the square root (then it would be easy to evaluate from there), but i cannot seem to figure it out (by completing the square or any other way)

    Thanks for any help
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by mybrohshi5 View Post
    I am having trouble solving this integral
     \int ^2_1 \sqrt(4x^2+4+\frac{1}{x^2})  dx

    I think i can reduce the equation under the square root to an expression that is squared which would get rid of the square root (then it would be easy to evaluate from there), but i cannot seem to figure it out (by completing the square or any other way)

    Thanks for any help
    4x^2+4+\frac{1}{x^2}=\left(2x+\frac{1}{x} \right)\left(2x+\frac{1}{x} \right)
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  3. #3
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    Hello, mybrohshi5!

    \displaystyle \int ^2_1 \sqrt{4x^2+4+\frac{1}{x^2}}\;dx

    Look at what's under the square root . . .

    . . 4x^2 + 4 + \dfrac{1}{x^2} \;=\;\dfrac{4x^4 + 4x^2 + 1}{x^2} \;=\;\dfrac{(2x^2+1)^2}{x^2}

    Or you can factor it like this: . 4x^2 + 4 + \dfrac{1}{x^2} \;=\;\left(2x + \dfrac{1}{x}\right)^2


    Got it?

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  4. #4
    Member mybrohshi5's Avatar
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    Oh boy i feel stupid that i missed that haha.

    Thanks for both of the fast replies
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