Why subsitution method for integration always work ?

[MIB]

why substitution method for integration always work ?

Why can we completely treat dx and du known in substitution method completely like differentials even if we don't have ∫f(g(x))g'(x) dx , i.e : why we can substitute x in terms of u and dx in terms of du .

thanks

 

[lurflurf]

The chain rule and implicit function theorem
if g is chosen as a function of x we have
f(g(x))g'(x) dx=f(g)dg
which comes from
dg=g'(x)dx
which is the chain rule precisely
if instead we chose an implicit relationship between g and x we have
h(g,x)=0
dh=h_xdx+h_gdg=0
dg=[dg/dx]dx=[-h_x/h_g]dx
which is the chain rule precisely