Let F(x) = integral(top is x, bottom is -2) 3t^2(cos(t^3) + 2) dt.
A. Using the First Fundamental Theorem of Calculus, find F'(x). (COMPLETED)
B. Starting with F(x)= integral(top is x, bottom is -2) 3t^2(cos(t^3)+2)dt, use the subsitution u(t)=t^3 to rewrite the definite integral. You should get a new (equivalent expression for F(x), which consists of this new definite integral.
C. Using your new expression for F(x) from part B and Leibniz's Rule, find F'(x).
F(x) = ∫[-2,x] 3t^2 (cos(t^3)+2) dt
F'(x) = 3x^2 (cos(x^3)+2)
if u = t^3 then
du = 3t^2 dt, making
F(u) = ∫[-8,x^3] (cos(u) + 2) du