Derivative inside integral of a translated function

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Hint: Given any integrable function $g$, $$\int_a^bg(x+y)dx=\int_{a+y}^{b+y}g(x)dx.$$

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Laura
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Laura

Updated on December 05, 2020

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  • Laura
    Laura almost 3 years

    The exercise state as follows: Let $f:R \rightarrow R$ be an absolutely continuous function over every compact in $R$. I'm asked to prove that: $\displaystyle \frac{d}{dy}\int_a^b f(x+y)dx=\int_a^b f'(x+y)dx $
    for every $y \in R$.

    I thought that if $f$ is absolutely continuous over the compacts subset of $R$ so is also the translation $f(x+y)$. I don't think I need to prove this. Do you think I should?

    I tried to apply the fundamental theorem but I didn't get anything useful: $ f(b)=f(a)+\int_a^b f'(x+y)dx$
    I also considered the derivative respect $x$ or $y$ the function $f'(x+y)$ is the same and it's equal to the derivative $f'(x)$. Is it right?
    But any of these idea seems to be helpful, I've also tried to apply the Lagrange theorem to the derivative and the function over the set $f(b)-f(a)$ but still no results.
    I hadn't tried with results on derivating under integral because the information on f and on its derivative seems definitely too weak in my opinion.So I don't know what to do now. Any idea?

  • Laura
    Laura almost 11 years
    I tried to develop your hint in the answer above.