Trying to prove that $\int f'(x)dx=f(x)$ gone wrong

Question

I was trying to prove that $$\int f'(x)dx=f(x)+C$$($f(x)$ is differentiable and $f'(x)$ is integrable) when I got this instead: $\int f'(x)dx=\int \frac{f(x)}{x}dx-f(0)\ln|x|+C.$
Here is how I got this: $$\int f'(x)dx=\int \lim_{h\rightarrow0}\frac{f(x)-f(h)}{x-h}dx=\int\lim_{h\rightarrow0}\frac{f(x)}{x-h}dx-\int\lim_{h\rightarrow0}\frac{f(h)}{x-h}dx=\int\frac{f(x)}{x}dx-\int\frac{f(0)}{x}dx=\int\frac{f(x)}{x}dx-f(0)\ln|x|.$$

I think I got this wrong when I split the limit, but it is convergent anyways. So where did I do this incorrectly?

Answer 1

First of all, we do not know anything about $f$ (continuity etc...). Second, the derivative is defined as $$ \lim_{h \rightarrow x } \frac{f(x)-f(h)}{x-h} $$ and not the way you defined it.

Also, rather try to prove $\frac{\mathrm{d}}{\mathrm{d}t} \int^t_a f(s)~\mathrm{d}s = f(t)$ for continuous $f$.