Use of substitution for integration

[Supernova123]

I was wondering if there is a convenient way of checking if a substitution is correct or not.
For example, I tried solving for ∫(1/(a^2-x^2)dx using two different substitutions, x=acosu and x=asinu giving different solutions. I got the integral as arcsin(x/a) using x=asinu and -arccos(x/a) using x=acosu. But which one of these is the correct one and how do I tackle this sort of problem if it arises again?

 

[dextercioby]

Differentiating back, of course. In your question, the 2 solutions are related through a constant, as they should be. Can you guess it? How about derive it?

 

[Stephen Tashi]

But which one of these is the correct one and how do I tackle this sort of problem if it arises again?

As dextercioby indicates, antiderivatives of a function are not unique. if the two answers differ by a constant then they could both be correct antiderivatives. (Consider things like $sin(\theta) = cos( \frac{\pi}{2} - \theta)$.)

 

[mathman]

arccos(x) + arcsin(x) = π/2.

 

[Supernova123]

Alright, thanks for the input. I'm guessing that since sin(x)=cos(π/2-x), then:
sin(x)=u, cos(π/2-x)=u
arcsin(u)=x, arccos(u)=π/2-x
So arcsin(u)+arccos(u)=x+π/2-x=π/2
Since they differ by a constant ,then
arcsin(x/a)+c=-arccos(x/a)
arcsin(x/a)+arccos(x/a)=c=π/2