Are the following equalities true?

Question

I have thought about the following equalities for a while and maybe you can suggest are they true or false, and help me to solve some inconsistencies which occur. I am making one question for three equalities because they are connected. So:
1) Is $$\int f(x)\,dx = b + \int f(x)\,dx $$ where $f$ is some function and $b$ is some number.
2) Is $$\int f(x)\,dx - \int f(x)\,dx = 0$$ where $f$ is some function. This equality may seem a bit obvious but in connection with other one it can cause some problems (inconsistencies).
3) I definitely know that: $$\int \frac{1}{x}\,dx = 1 + \int \frac{1}{x}\,dx$$ and I know this because you can easily prove it using integration by parts. And now, this equality in combination with (2) leads to the following false equality $$0=1$$ So there can be two posibilities: either (2) is false or I am doing some technical mistakes when I conclude that $$0=1$$ So what is the mistake ? Thanks in advance

Answer 1

The problem arises at the notation of $\int f(x)dx$. On the first sight, it seem that you get a unique function $F(x)=\int f(x)~dx$, but that is false. There are infinte many primitives for a function $f$ which are equation up to a constant. If you don't need to specify the constant, then you can use the notation $\int f(x)~dx$. But if you write down $$ \int f(x)dx = b+\int f(x)dx, $$ then you can indeed find primitives on the left and right side such that the equation holds, but then the same term has different meaning in one equation and you can't just say that they has to be the same because of the same notation.

Therefore, you should use a better notation like $$ F(x)=\int_a^x f(t)~dt. $$ Here, $F$ is a specific primitive of $f$ and now $$ F(x)=b+F(x) $$ implies $b=0$.

The same problem apears in 3). The equation is true in the sense that there exists different primitives on the LHS and RHS of the equation such that the equation holds. But further the primitive on the LHS and RHS are not the same and you can't just drop it.

Finally, 2) has the same problem. If you use the same primitive for both integrals, the equation is true.