Find all functions that satisfy that $\int_{0}^{x}f(t)dt = e^{x}$.

Question

I used the Fundamental Theorem of Calculus and I obtained that $f(x) = e^{x}$ but when I integrate f, it results in $e^{x} - 1$.

There are no functions which satisfy $\int_{0}^{x}f(t)dt = e^{x}$. However, a very common definition of $\text{exp}(x)$ is the unique function such that $f′(x)=f(x)$ and $f(0)=1$ as shown here. — Axion004, Mar 23 '20 at 20:30

score 10 · Accepted Answer · answered Mar 23 '20 at 20:12

10

There are no such functions. Note that taking $x=0$ on the left gives $0$ while on the right it gives $1$.

answered Mar 23 '20 at 20:12

lulu

Why FTC doesn't apply? – Salihcyilmaz Mar 23 '20 at 21:06
2

@Salihcyilmaz because the initial equation is internally contradictory. If you start from a false premise you can deduce whatever you want. – lulu Mar 23 '20 at 21:09
Ah, i see now, thanks. – Salihcyilmaz Mar 23 '20 at 21:11

1 Answers1