What happens if one of the boundaries in the integral is a function of x, fundamental theorem of calculus?

If one of the boundaries in an integral is a function of x, the fundamental theorem of calculus still applies. you just end up using the chain rule if you differentiate the integral.for example, consider the following integral:       if we let the anti-derivative of the integrand (t cos t) be represented by a function f, then f'(t) = t cos t. by the fundamental theorem of calculus, the integral can be rewritten as follows:     if we were to differentiate this integral expression with respect to x, we would have to use the chain rule    since f'(t) = t cos t, then f'(x³) = x³ cos x³ (just normal chain rule procedure)    ======here's another example