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I don't understand the following:

[tex]\frac{d^{2}}{dt^{2}}\int_{0}^{t}(t-\epsilon )\phi (\epsilon)d\epsilon=\phi''(t)[/tex]

All I know is:

[tex]\frac{d^{2}}{dt^{2}}\int_{0}^{t}(t-\epsilon )\phi (\epsilon)d\epsilon=\frac{d^{2}}{dt^{2}}\int_{0}^{t}t \cdot \phi (\epsilon)d\epsilon-\frac{d^{2}}{dt^{2}}\int_{0}^{t}\epsilon \cdot \phi (\epsilon)d\epsilon[/tex]

Is it allowed to say:

[tex]\frac{d^{2}}{dt^{2}}\int_{0}^{t}t \cdot \phi (\epsilon)d\epsilon=\frac{d^{2}}{dt^{2}}\cdot t\int_{0}^{t}\phi (\epsilon)d\epsilon[/tex]?

And if so, why is this correct? This is only correct when [tex]t\neq f(\epsilon )[/tex], right? But I am not sure whether this is the case...

Thank you in advance!

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# Second derivative of an integral

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