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I did an exercise which would like to calculate the derivative in $x$ of the integral $$T(x) = \int\limits_0^{1} t^2 (1-t)^3 (t-x)^4 dt,$$ Of course, I showed that $$\dfrac{d}{dx} T(x) = - 4 \int\limits_0^{1} t^2 (1-t)^3 (t-x)^3 dt . $$

The problem is that I'd like a general resultat which can give a method to determinate the derivative in $x$ of the integral $T_1(x) = \int\limits^1_0 f(t, x) dt$ where $f(t,x) $ is a polynomial function of $t$ and $x$.

My question: Is there a resultat to determinate the derivative in $x$ of the integral $T_1(x) = \int\limits^1_0 f(t, x) dt$ where $f(t,x) $ is a polynomial function of $t$ and $x$?

Thank you very much for your interests!

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    $\begingroup$ Have you learned about partial derivatives? You can do this by Leibniz's rule for differentiation under the integral sign. $\endgroup$ – saulspatz Apr 4 at 2:25
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    $\begingroup$ Multiply out the polynomials. You should get the same result regardless of whether you differentiate with respect to $x$ before or after you integrate. $\endgroup$ – Doug M Apr 4 at 2:38
  • $\begingroup$ @saulspatz Yes, it is nice! Thank you so much! $\endgroup$ – mathJuan Apr 4 at 2:39
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In general, if $a$ and $b$ are constants, and $$F(x)=\int_a^b f(x,t)dt$$ then $$F'(x)=\int_a^b\frac{\partial}{\partial x}f(x,t)dt$$ for a sufficient function $f$. Here $\frac{\partial}{\partial x}$ denotes the partial derivative. More details can be found here.

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