$\int^x_0 f (t) $dt =F(x) for every ∈ [0,2]

Question

$\int^x_0 f (t) $dt =F(x) for every ∈ [0,2]

I need to find all x where : F'(x) = f(x)

1. $$ f(x) = \left\{ \begin{array}{c} 0, \ ≤ 1 \\ 1 , > 1 \end{array} \right. $$

2. $$ f(x) = \left\{ \begin{array}{c} 1, \ = 1/n \\ 0 , ≠ 1/n \end{array} \right. $$

3. $$ f(x) = \left\{ \begin{array}{c} 1, \ ≠ 0 \\ 1/ [ 1/ ] , = 0 \end{array} \right. $$

Answer 1

I show you 1.:

If $0 \le x \le 1$ we have $F(x)= \int_0^x 0 dx=0$.

If $1 <x \le 2$ we have $F(x)= \int_0^1 0 dx+ \int_1^x 1 dx=x-1$.

Thus:

$$F(x) = \left\{ \begin{array}{c} 0, \ 0 \le ≤ 1 \\ x-1 , 1<x \le 2 \end{array} \right..$$