FTC with a two-variable function

[e^(i Pi)+1=0]

https://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign

I don't understand where the last term comes from, the one that's an integral of a partial derivative. When I solve it using the FTC I get the same answer minus that term.

If I differentiate first then integrate I get that term but then none of the others.

 

[Ray Vickson]

https://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign

I don't understand where the last term comes from, the one that's an integral of a partial derivative. When I solve it using the FTC I get the same answer minus that term.

If I differentiate first then integrate I get that term but then none of the others.

Work it out for yourself from first principles: if
$$F(x) = \int_{a(x)}^{b(x)} f(t,x) \, dt,$$
then
$$\frac{d}{dx} F(x) = \lim_{h \to 0} \frac{F(x+h) - F(x)}{h}$$
Work out the value of the numerator for small $h > 0$; you will see that in general it involves several types of terms, and these give limits that are the terms in the final result you want.