How does the Leibniz integral rule apply when differentiating under the integral sign?

[filter54321]

I'm in real analysis classes but my calculus is shaky when I hit things that aren't plug and chug. How do you evaluate this integral and why can you ignore the dx differential?

The more theoretical details, it smells like some flavor of the Fundamental Theorem of Calculus or Leibniz's Rule but I'm lost:

Differentiate with respect to t:

d/dt [ t + Integral(m(t-x) dx, 0, t) ]

It's supposed to evaluate to 1 + m(t). I see where the 1 comes from but I don't know anything about m(t) though so there has to be a general principle...


Thanks

 

[ptolema]

change the limits on the integral. let g(x)=t-x, and change the limits to g(0) and g(t) before deriving

http://www.cliffsnotes.com/study_guide/Definite-Integrals.topicArticleId-39909,articleId-39903.html"
go to the definite integral evaluation section for more detail

 

[vela]

You're right. It's an application of the http://en.wikipedia.org/wiki/Leibniz_integral_rule#Variable_limits". Where are you getting stuck?