Proving a property of an integral

[Mike s]

I have already solved it, but I need confirmation:

Are there other ways of proving this?

Thanks in advance!

 

[Curious3141]

I have already solved it, but I need confirmation:

Are there other ways of proving this?

Thanks in advance!

Your proof is fine (and it's the way I would've done it), except that you should explicitly define your $F(a)$. You implicitly defined it as an indefinite integral, which means $F(0) = c$, but I would prefer to define $F(a) = \int_0^a f(x) dx$, and include one more intermediary step clarifying that $\int_a^{2a} f(t) dt = \int_0^{2a} f(t) dt - \int_0^a f(t) dt = F(2a) - F(a)$. This way, I don't have to bother with the $F(0)$ term at all.

 

[Mike s]

Your proof is fine (and it's the way I would've done it), except that you should explicitly define your $F(a)$. You implicitly defined it as an indefinite integral, which means $F(0) = c$, but I would prefer to define $F(a) = \int_0^a f(x) dx$, and include one more intermediary step clarifying that $\int_a^{2a} f(t) dt = \int_0^{2a} f(t) dt - \int_0^a f(t) dt = F(2a) - F(a)$. This way, I don't have to bother with the $F(0)$ term at all.

Thanks a lot!