Question

Can someone explain what I wrote down in the purple circle? I wrote it down when my teacher was talking but I don’t remember what it means.

Answer 1

Explanation:

This is the second fundamental theorem of calculus.

d/dx ∫ₐᵇ f(x) dx = f(b) (db/dx) − f(a) (da/dx)

This is derived using chain rule:

d/dx g(f(x)) = g'(f(x)) f'(x)

Therefore:

d/dx ∫₀²ˣ arctan(t) dt = arctan(2x) (2x)' = 2 arctan(2x)