Substitution formula for integrals

[aaaa202]

I suppose you all know the substitution formula for integrals.

Well sometimes it seems to me you use substitutions which just don't fit directly into that formula.

For instance for the integral of 1/(1+x^2) you substitute x=tan(u). Why is it suddenly allowed to assume that x can be expressed through the tangent function? - certainly it just can't the the good old formula for integration by substitution?

 

[rock.freak667]

Those substitutions are used to make the integrals easier. In this case if x=tanu, then the denominator will become 1+tan^2(u) which is the same as sec^2(u) which is easier to deal with.

But if you know your inverse trig differentials then you could see that d/dx(tan^-1x) = 1/(1+x^2)

so putting x=tanu is not so farfetched.