I was Reading a book about calculus when I've found this part about variable substitution in integrals:
Consider $f$ defined in na interval $I$. Suppose that $x =\phi(u)$ is inversible, and its inverse is $u = \theta(x), x \in I$, $\phi$ and $\theta$ are differentiable.
Then the book throws this integral: $$\int f(\phi(u))\phi'(u)du = F(u) + k \ \ (u \in D_{\phi})$$
Then
$$\int f(x)dx = F(\theta(x)) + k$$
As far as I know, the integral $\int f(\phi(u))\phi'(u)du$ should be equal $F(\phi(u))$ by the reverse chain rule method, so $F(u)$ does not makes sense to me. Please could somebody explain to me what's happening? I'm really really confused. My head is exploding.