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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.

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  • $\begingroup$ Is it saying if the first equation is true, then the second is true? $\endgroup$ – Michael Albanese Jun 6 '14 at 7:31
  • $\begingroup$ @MichaelAlbanese I don't know, the book simply says this I typed, but is in portuguese, so I translated. $\endgroup$ – Marter Js Jun 6 '14 at 15:26

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