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How would one interpret: $\frac{\mathrm d}{\mathrm dx}\int_0^x (F(y)-F(x))\,\mathrm dy$ I don't think I can use the fundamental theorem of calculus here, can I?

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    @J.M. Thanks for the feedback. I suppose I am a little quick on accepting in general.2011-10-08

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Certainly doable:

$\begin{split}\frac{\mathrm d}{\mathrm dx}\int_0^x (F(y)-F(x))\,\mathrm dy&=\frac{\mathrm d}{\mathrm dx}\left(\int_0^x F(y)\mathrm dy-F(x)\int_0^x \,\mathrm dy\right)\\&=\frac{\mathrm d}{\mathrm dx}\left(\int_0^x F(y)\mathrm dy-x\,F(x)\right)\\&=F(x)-x\,F^\prime(x)-F(x)\\&=-x\,F^\prime (x)\end{split}$

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    @HenningMakholm under what conditions it is going to hold? I am solving a similar problem, but the bounds of integration are $(-\infty, \infty)$2014-05-05