-
$\int_a^x f(t)dt = F(t)\Big|_a^x = F(x) - F(a)$
-
$ \frac{d}{dx}\int_a^x f(t)dt = f(x)$
-
$ \frac{d}{dx}\int_x^b f(t)dt = -f(x)$
-
$ \frac{d}{dx}\int_{u(x)}^{v(x)} f(t)dt = f(v(x))\frac{dv}{dx} -f( u(x)) \frac{du}{dx} $
-
$ \frac{d}{dx}\int_{u(x)}^{v(x)} f(x,t)dt = f(x,v(x))\frac{dv}{dx} -f(x, u(x)) \frac{du}{dx} + \int_{u(x)}^{v(x)} \frac{\p f}{\p x}(x,t)dt $