I'm reading Zorich, Mathematical Analysis I, and I found a not clear step in the paragraph on Primitives. The particular sentence is shown below (adapted).
From the definition of primitive of a function on an interval, and from the properties of differentiation, the following relation holds: $$\int (f\circ \phi)(t)\phi '(t)\mathrm{d}t=(F\circ \phi)(t)+c$$ where $F$ is the primitive of $f$ on an interval $I\subset\mathbb{R}$, and $\phi:I_t\subset\mathbb{R} \to I_x\subset I$ is a function $C^{(1)}(I_t)$.
I can not understand why it is necessary that $\phi '(t)$ has to be continuous in $I_t$, what would be wrong if it were not? I can not see the problem in that case.
Thanks.
Assuming the notation $\int g(t)\,dt = G(t), t\in I,$ means simply that $G'(t)=g(t),t\in I,$ you are correct. You might have problems with
$$\int_a^b (f\circ \phi)(t)\phi'(t)\,dt = (F\circ \phi)(b)-(F\circ \phi)(a)),$$ however. Now that I think about it, you could have problems with that even with Zorich's formulation.