Taking the derivative inside the integral (Liebniz Rule for differentiation under the integral sign)
Solution 1:
Yes, you can, assuming some weak conditions are met. If $h(x,t)$ has continuous partial derivatives, then $$\frac{\mathrm{d}}{\mathrm{d}x} \left (\int_{a}^{b}h(x,t)\,\mathrm{d}t \right) = \int_{a}^{b} \frac{\partial}{\partial x}h(x,t)\; \mathrm{d}t.$$