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If $f(x)$ is continuous, then the reference answer to the derivative of this definite integral $\frac{d}{d x} \int_{0}^{x} t f\left(x^{2}-t^{2}\right) d t$ is $x f\left(x^{2}\right)$.

FullSimplify[D[Integrate[t*f[x^2 - t^2], {t, 0, x}], x]]

But the above code can not directly get the result form of the reference answer. What can I do to get the result form I want directly?

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    $\begingroup$ Use Mathematica 8 if you want the result directly. Later versions will not do derivatives of definite integrals because Mathematica does not know f is continuous, and I know of no way to tell it that f is continuous. With current versions you can take derivatives of indefinite integrals and apply the limits yourself. $\endgroup$
    – Bill Watts
    Aug 23, 2020 at 6:02
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    $\begingroup$ @BillWatts In V12.1.1, Mathematica takes the derivative assuming f is smooth, but it won't do integration by parts on a definite integral with the symbolic function f'[x]. $\endgroup$
    – Michael E2
    Aug 23, 2020 at 16:16

1 Answer 1

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D[Integrate[t*f[x^2 - t^2], {t, 0, x}], x] == Integrate[D[t*f[x^2 - t^2], x], {t, 0, x}]

is solvable with Mathematica since Integrate can deal with formal integrals if the integral in infinite.

As the short passage about formal integrals in the documentation page shows, prefer integrals without bounds.

Integrate[2 t x Derivative[1][f][-t^2 + x^2], t]

(-x f[-t^2 + x^2])

By hand this can be set into bounds t==0 to t==x:

-xf[0]+xf[x^2]

This problem does not depend on the symbol for the upper bound. This is done with Mathematica V12.0.0.

There is the condition necessary that f[0]==0 to get Your result.

To change the sequence order between integration and differentiation there is another condition necessary: f has to be sufficiently steady and the derivate has to exist on as much points as the function is steady on.

Reasons for that are given in the section Possible Issues for Indefinite Integrals and Definite Integrals. Example:

When part of a sum cannot be integrated explicitly, the whole sum will stay unintegrated:

Integrate[f[x] + f'[x], x]

([Integral](f[x] + Derivative[1][f][x]) [DifferentialD]x)

If entered different:

Integrate[f[x], x] + Integrate[f'[x], x]

(f[x] + [Integral]f[x] [DifferentialD]x)

There is a Mathematica built-in called Derivative that can do derivative with negative integer order:

Derivative[-1][Function[t, 2 x t f'[-t^2 + x^2]]]

(Function[x, -x f[-t^2 + x^2]])

Both together stays unevaluated in the abstract f situation.

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