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$\frac{d}{dq}\int_{s_{1}-z-q}^{z-s_{1}} \varphi(w) \, dw$

(if it helps, in my setting $\varphi$ is the CDF of some arbitrary uniform distribution). So I want to get a nice expression for this integral and it seems to suggest FTC, but I tried a change of variable and ended up with a $q$ inside the integrand which was not nice. Any help much appreciated.

2 Answers 2

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Use the chain rule:

$\frac{d}{dq}\int_{s_{1}-z-q}^{z-s_{1}} \varphi(w) \, dw = -\frac{d}{dq}\int_{z-s_1}^{s_1-z-q} \varphi(w) \, dw$

$ = - \frac{d}{d(s_1-z-q)} \int_{z-s_1}^{s_1-z-q} \varphi(w) \, dw \cdot\frac{d}{dq}(s_1-z-q)$

$ -\varphi(s_1-z-q)\cdot (-1). $

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Use the Leibnitz Rule of Differentiation of Integrals in the case that the limits of integration depend on the differentiation variable.

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    Though perhaps too powerful, it does indeed solve the problem. Sometimes one cares less about the mode of transportation than the destination.... ;) Plus, it isn't clear that $w$ isn't a function of $q$ from the OP.2012-04-22