How is it possible to prove that the indefinite integrals of "equivalent" functions that cannot be defined at a particular point are equal to each other?
For example,
$$\int\dfrac {x^2}{x}dx=\int xdx $$
Or,
$$\int\dfrac {(x+1)^2}{x+1}dx=\int (x+1)dx $$
Obviously, the function $f(x)=\dfrac{x^2}{x}$ is not defined at the point $x = 0$, but $f(x)=x$ is defined and likewise, the function $f(x)=\dfrac {(x+1)^2}{x+1}$ is not defined at the point $x = -1$, but $f(x)=x+1$ is defined.
How can I prove that, the indefinite integrals of these functions must be equal?
Thank you very much.
When you are solving the indefinite integral you are looking for primitive functions (of the sub-integral function) in some interval $I$. You need to have an idea up-front what this interval is.
Now... if this interval includes the point(s) -1/0 the sub-integral function is not defined at that point, so really it makes no sense to look for a primitive function at that point (for one or both of the functions). You see if a function is not defined at a given point, it has no primitive too.
If on the other hand the interval does not include the point(s) -1/0, then of course you can cancel the sub-integral function/fraction and solve. So then the primitive function is if of course the same (plus some constant $C$).
So in general the two indefinite integrals are not quite the same.