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!