Find the derivative using...

Question

Find the derivative of the following function

𝐹(𝑥)= ∫𝑥^3 (upper) 𝑥^2(lower) (2𝑡−1)^3𝑑𝑡

using the Fundamental Theorem of Calculus. 

𝐹′(𝑥) = ?

Answer 1

To take the derivative of an anti-derivative, drop the integral sign and the dt, and plug in your bounds for the expression in the integral.

Let's write f(t) = (2t-1)³. The fundamental theorem of calculus tells us that F'(x) = f(x³) - f(x²).

Answer 2

F(x) = ∫_{(from x squared to x cubed)} (2t - 1)³dt =

∫_{(from x squared to 0)} (2t - 1)³dt + ∫_{(from 0 to x cubed)} (2t - 1)³dt =

-∫_{(from 0 to x squared)} (2t - 1)³dt + ∫_{(from 0 to x cubed)} (2t - 1)³dt

So, F'(x) = -(2x² - 1)³ (x²)' + (2x³ - 1)³ (x³)' = -2x(2x² - 1)³ + 3x²(2x³ - 1)³