Why subsitution method for integration always work ?

  • Thread starter MIB
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  • #1
MIB
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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
 
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Answers and Replies

  • #2
lurflurf
Homework Helper
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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=hxdx+hgdg=0
dg=[dg/dx]dx=[-hx/hg]dx
which is the chain rule precisely
 

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