rules for constant resulted from integration.

Question

$$ \int y'dx $$ y' is first derivative and function of x so integration cancels derivative sign. As a result we get from above operation.

$$ = y + C $$

My question is if C is an arbitrary constant then, can I chose 2C instead of C ? as in

$$ = y + 2C $$

Thank you.

Answer 1

Yes. The integral sign the way you use it (i.e. without lower and upper bound) just means "antiderivative" or "indefinite integral". That is to say $$ \int f(x) dx = F(x) :\Leftrightarrow F'(x)=f(x) \forall x\in D$$ where $D$ is some domain. For your question: You are right, since $$ \frac{d}{dx} \big(y(x)+2C\big)= y'(x)$$ as long as $C$ is a constant wrt to $x$.

Answer 2

Yes, because $(y+2C)'=y'$ and $(y+C)'=y'$, so both $y+2C$ and $y+C$ are primitives of $y'$, by the definition of indefinite integrals, both go through.