a question on indefinite integral

Question

I have a question. Suppose the function f(a,x) has good enough properties. Then does the following equality holds?

$\frac{\partial \int f(a,x) dx}{\partial a}=\int \frac{\partial f(a,x)}{\partial a}dx +C_0$

Thanks ahead.

Answer 1

Consider the following example, i.e. the convolution integral:

$F(t)=\int_{a}^{t}f(\tau)g(t-\tau)d\tau$, $a$ being constant. Define $h(t,\tau)=f(\tau)g(t-\tau)$. Assume $h$ be continuous in a closed interval $I$. Then, it can be proven as generalisation of the Foundamental Theorem of Calculus:

$\dot{F}(t)=h(t,\tau)_{|\tau=t}+\int_{a}^{t}\dot{h}(\tau,t)d\tau=f(t)g(0)+\int_{a}^{t}f(\tau)\dot{g}(t-\tau)d\tau$. (Eq. 1)

(where $\dot{f}$ refers to the first-order time derivative of $f$ w.r.t. $t$). In other words, if the function to be integrated depends on any of the extremes of integration, the integral at right-hand side of (Eq. 1) shows up. Be careful: you didn't specify the extremes of integration and your equality is generally false (since $C_0$ is generally a function). However you can take the formula used in this example as a general reference.