I have a theorem in my notes that states:
Let $f$ be an injective and differentiable function in an open interval $I$. We can show that $J=g(I)$ is an open interval, and that $f^{-1}$ is differentiable in $J$. Show that $$\int_{a}^{b}f(x)dx+\int_{f(a)}^{f(b)}f^{-1}(y)dy=bf(b)-af(a).$$
Now, reading this version of the theorem I noticed it used as hypothesis that $f$ is continuous and invertible. What I don't know is if the injectivity of $f$ is enough to prove the theorem. Isn't it true that a function is bijective if and only if it has an inverse? How is injectivity and differentiablity useful here?
Also, I was able to algebraically prove the equality above.
To answer your first question, yes, a function is bijective if and only if it has an inverse. You can find a detailed proof in Herno's answer to this question. Then injectivity for $f$ is necessary for the result to hold as $f$ needs to be invertible, then bijective, so in particular injective.
If you look at the proof without words in the Wikipedia page you linked, it becomes apparent that if $f$ and $f^{-1}$ are integrable functions then the result should hold (look in particular at this image). In this case, continuity for $f$ and $f^{-1}$ suffices to guarantee (Riemann) integrability, so it can be proved without using the stronger hypothesis of differentiability and replacing it by continuity for $f$ (with the necessary injectivity). Of course, differentiability allows the use of stronger theorems and results than just continuity, so some different proofs may be enabled once one assumes the functions $f$ and $f^{-1}$ are differentiable (as is mentioned in the article, differentiating will prove the result immediately).