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Thread: How would you guys prove this integral?

  1. #1
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    How would you guys prove this integral?

    How would you guys prove this integral?-capture.png
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  2. #2
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    Re: How would you guys prove this integral?

    Use the definition of an even function.
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  3. #3
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    Re: How would you guys prove this integral?

    I have always found this to be very tricky
    $ \begin{align*}\int_{ - 2}^0 {f(x)dx} &= - \int_{ 2}^0 {f( - u)du}\text{ What was done here?}\\ &= \int_{ 0}^2 {f( - u)du}\text{ What was done here and why?}\\&=\int_{ 0}^2 {f( u)du}\text{ Where is evenness used?}\\&=\int_{ 0}^2 {f( x)dx}\text{ Be careful in explaining this step!} \text{ ***} \end{align*}$



    $ \begin{align*} \int_{ -2}^2 {f( x)dx}&=\int_{ -2}^0 {f( x)dx}+\int_{ 0}^2 {f( x)dx} \\&=\int_{ 0}^2 {f( x)dx}+\int_{ 0}^2 {f( x)dx}\\&=2\int_{ 0}^2 {f( x)dx} \end{align*}$.
    Thanks from lc99
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    Re: How would you guys prove this integral?

    that was very clever the way that worked!
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