I need step by step help to find the derivative of a problem, using FTC part 1 please.

Question

g(x) =∫_x^{x^2} (2t² + 3) dt

Answer 1

Since both limits of integration are variable, we need to split the integral into two integrals

g(x) = ∫⁰_xh(t)dt + ∫₀^{x^2}h(t)dt where h(t) = 2t² + 3 in this case

Swapping the limits on the first half makes it negative.

With functions as limits, is like having a function F(g(x)) and d/dx(F(g(x)) = F'(g(x))g'(x) due to the chain rule

so, in general

∫^v(x)_u(x) h(t)dt = h(v(x))v'(x) - h(u(x))u'(x)

g'(x) = (2(x²)²+3)(2x) - (2x²+3)(1)

= 2x⁵+6x - 2x²-3