FTC — The Bridge Between Derivatives & Integrals

FTC Part 1 (Evaluation):

∫ from a to b of f(x)dx = F(b) - F(a)

Find the antiderivative, evaluate at endpoints, subtract.

FTC Part 2 (Derivative of an integral):

d/dx[∫ from a to x of f(t)dt] = f(x)

The derivative of an accumulation function equals the integrand evaluated at the upper limit.

With chain rule:

d/dx[∫ from a to g(x) of f(t)dt] = f(g(x)) · g'(x)

Example: d/dx[∫ from 0 to x² of sin(t)dt] = sin(x²) · 2x

FTC Part 2 is tested on the AP exam every single year. Usually as a multiple choice question. Know the chain rule version.

Reference:

Wikipedia: FTC

https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus