1 Leibniz Integral Rule
1.1 Leibniz Integral Rule
In calculus the Leibniz integral rule for differentiation under the integral sign states that for an integral of the form \[ \int_{a(x)}^{b(x)}{f(x,t)}~d{t}, \] where \(\lvert a(x) \rvert, \lvert b(x) \rvert< \infty\) and the integrands are functions dependent on \(x\), the derivative of this integral is expressible as \[ \frac{d}{dx}\left( \int_{a(x)}^{b(x)}{f(x,t)}~d{t} \right) =f(x,b(x))\cdot \frac{d}{dx}b(x)-f(x,a(x))\cdot \frac{d}{dx}a(x)+\int_{a(x)}^{b(x)}{\frac{\partial}{\partial x}f(x,t)}~d{t}, \] where the partial derivative \(\frac{\partial}{\partial x}\) indicates that inside the integral, only the variation of \(f(x,t)\) with \(x\) is considered in taking the derivative.
Example: