What are real functions f() if f'(x)=2x+1 and f(0)=0?

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Given problem is

`f'(x)=2x+1` and `f(0)=0`

Let us assume that `f'(x)` is well behaved i.e. it is integrable

Integrate

`intf'(x)dx=f(x)`

so

`f(x)=int(2x+1)dx +c`

`f(x)=2x^2/2+x+c`

`f(x)=x^2+x+c` ,where c is integrating constant

By given condition

`f(0)=0=0^2+0+c`

C=0

Thus real function is

`f(x)=x^2+x`

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