Question about the substitution rule. Where did I go wrong?

Question

$$\int (f(x))' dx = f(x) + c$$

if $u=g(x)$ then

$$\int (f(u))'du = f(u)+c$$

But

$$\int (f(g(x)))'dx = f(g(x))+c$$

Where did I go wrong?

Answer 1

In short, you forgot the chain-rule. That is to say that if $u = g(x)$, then $du = g'(x)dx$. It is this that let's us say that

$$ \int [f(g(x))]'dx =\int f'(g(x))g'(x)dx = f(g(x)) + C$$