If we have a map $f: X \to Y$ and $\omega$ is a differential form on $Y$ with same dimension as $X$. When can we have the following identity, $$\int_X f^*\omega=\int_{f(X)}\omega$$ I only know that when $f$ is an orientation-preserving diffeomorphism onto its image, this is true. Are there any other wider cases for this to hold?
Also, when can we have it's valid on topological level (under the assumption that $\omega$ and $X$ are both closed, that is $$\langle f^*[\omega],[X]\rangle = \langle [\omega],f_*[X]\rangle$$