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Thread: How to solve this integral using substitution?

  1. #1
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    How to solve this integral using substitution?

    $$int{(x-(\sqrt{x}-5)^2)^2} dx$$
    I find the answer using expansion. How to solve it using substitution?
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  2. #2
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    Re: How to solve this integral using substitution?

    To me an obvious first step is to let $u= \sqrt{x}= x^{1/2}$. Then $x= u^2$ and $dx= 2udu$.

    The integral becomes $2\int [u^2- (u- 5)^2](u du)= 2\int (10u- 25)udu= 10\int 2u^2- 5u du$.
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  3. #3
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    Re: How to solve this integral using substitution?

    It will be $$\frac{20}{3}u^3 - \frac{50}{2}u^2$$
    But it still wrong..
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  4. #4
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    Re: How to solve this integral using substitution?

    Quote Originally Posted by Helly123 View Post
    It will be $$\frac{20}{3}u^3 - \frac{50}{2}u^2$$
    But it still wrong..
    Go ahead and use $u=\sqrt{x}$.
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  5. #5
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    Re: How to solve this integral using substitution?

    Quote Originally Posted by Helly123 View Post
    It will be $$\frac{20}{3}u^3 - \frac{50}{2}u^2$$
    But it still wrong..
    That is because of a small typo that HallsofIvy made. His method is spot on.

    $\displaystyle \begin{align*}\int \big( x- (\sqrt{x}-5)^2 \big)^2 dx & = 2\int \big(u^2-(u-5)^2 \big)^2(udu) \\ & = 2\int (10u-25)^2udu \\ & = 50\int (4u^3-20u^2+25u)du \\ & = 50\left(u^4 - \dfrac{20}{3}u^3+\dfrac{25}{2}u^2\right) + C \\ & = 50x^2-\dfrac{1000}{3}x^{3/2} + 625x + C\end{align*}$

    Halls just missed an exponent of 2.
    Thanks from topsquark and HallsofIvy
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  6. #6
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    Re: How to solve this integral using substitution?

    Ok thank you so much
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