Why subsitution method for integration always work ?

In summary, the substitution method for integration always works because it follows the chain rule and implicit function theorem. This allows us to treat dx and du as differentials and substitute x in terms of u and dx in terms of du.
  • #1
MIB
17
0
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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  • #2
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
 

1. Why is substitution method used for integration?

The substitution method is used for integration because it allows for the integration of more complicated functions that cannot be easily integrated using traditional methods. It simplifies the integration process by substituting a variable or expression with a new variable, making it easier to integrate.

2. How does substitution method work for integration?

The substitution method works by substituting a variable or expression in the integrand with a new variable. This new variable is chosen in such a way that it simplifies the integrand and makes it easier to integrate. The resulting integral is then solved using traditional integration techniques.

3. Is substitution method always applicable for integration?

No, substitution method is not always applicable for integration. It can only be used when the integrand contains a function that can be substituted with a new variable. If the integrand does not contain such a function, then substitution method cannot be used.

4. What are the benefits of using substitution method for integration?

The benefits of using substitution method for integration include simplifying the integrand, making it easier to integrate, and allowing for the integration of more complicated functions. It also helps to avoid using more complex integration techniques, such as integration by parts.

5. Are there any limitations to using substitution method for integration?

Yes, there are limitations to using substitution method for integration. It may not be applicable for all integrands, and it may not always lead to a simpler integral. In some cases, substitution method may make the integral more complicated and not easier to solve. It also requires a good understanding of substitution and the choice of the new variable can be challenging in some cases.

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