Use the second fundamental theorem of calculus to find F'(x)
F(x)=The integral from 0 -> x^2
(Sin(x)^2)dx
sin^2(x^2) * 2x
To find F'(x) using the second fundamental theorem of calculus, we need to differentiate the integral function F(x) with respect to x.
The second fundamental theorem of calculus states that if F(x) = ∫[a to x] f(t) dt, where "f(t)" is a continuous function, then F'(x) = f(x).
In this case, we are given F(x) = ∫[0 to x^2] (sin(x)^2) dx. We can differentiate F(x) by applying the fundamental theorem as follows:
Step 1: Rewrite the integral with limits.
F(x) = ∫[0 to x^2] (sin(x)^2) dx
Step 2: Define a new function.
Let G(t) = ∫[0 to t] (sin(x)^2) dx.
Step 3: Differentiate G(t) to find G'(t).
By applying the second fundamental theorem of calculus, we know that G'(t) = (sin(t)^2).
Step 4: Substitute x^2 for t.
Since F(x) = G(x^2), we can differentiate F(x) by substituting x^2 for t in G'(t).
Therefore, F'(x) = (sin(x^2)^2).
So, F'(x) = (sin(x^2)^2) is the derivative of F(x) using the second fundamental theorem of calculus.