Definite Integral of Partial Derivative

Theorem
Let $\map f {x, y}$ and $\map {\dfrac {\partial f} {\partial x} } {x, y}$ be continuous functions of $x$ and $y$ on $D = \closedint {x_1} {x_2} \times \closedint a b$.

Then:
 * $\ds \frac \d {\d x} \int_a^b \map f {x, y} \rd y = \int_a^b \map {\frac {\partial f} {\partial x} } {x, y} \rd y$

for $x \in \closedint {x_1} {x_2}$.