Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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: $$\int f(\phi(u))\phi'(u)du = F(u) + k \ \ (u \in D_{\phi})$$


$$\int f(x)dx = F(\theta(x)) + k$$

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.

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.