Is there a general result for the integral of F(G(X)) when F(x) is a x^k

[Al-Layth]

hi

I know ive been posting a lot of questions
Sorry about that if its a problem.

i am aware there is no general result for an integral of F(G(x)) So I am trying to find out more specific results

Question: Is there a general result for the integral of F(G(X)) when F(x) is x^k

Random example
(1+x^10)^0.5

F(x)= x^0.5
G(x) = 1+x^10

 

[Deleted member 4993]

hi

I know ive been posting a lot of questions
Sorry about that if its a problem.

i am aware there is no general result for an integral of F(G(x)) So I am trying to find out more specific results

Question: Is there a general result for the integral of F(G(X)) when F(x) is x^k

Random example
(1+x^10)^0.5

F(x)= x^0.5
G(x) = 1+x^10

Unless you define your F(x) and G(x) - like you did in this post (op) - I do not know of any rule like that.

I think you are looking for inverse of chain-rule of differentiation [d/dx (F(G(x) = F' * G + G' * F ]. As far as I know, that is non-existent.

 

[Al-Layth]

Unless you define your F(x) and G(x) - like you did in this post (op) - I do not know of any rule like that.

I think you are looking for inverse of chain-rule of differentiation [d/dx (F(G(x) = F' * G + G' * F ]. As far as I know, that is non-existent.

What if F(x) is specifically x^k
and you compose it with G(x) [ a general function ]
is there an inverse of the chain rule for this case?

 

[Dr.Peterson]

What if F(x) is specifically x^k
and you compose it with G(x) [ a general function ]
is there an inverse of the chain rule for this case?

The inverse of the chain rule is the substitution rule:

Integration by substitution - Wikipedia

en.wikipedia.org

There is no simple rule for the integral of a product or of a composition or of an inverse function, as there is for differentiation. Instead, you have to recognize some pattern in the integrand, rather than just following a routine. And many integrals just can't be done at all, whereas any combination of basic functions can be differentiated.

Generally, integration is more complex than differentiation, and I don't expect any rules other than the basic ones that you learn. (Integration by parts is the inverse of the product rule.) I compare it to division, the inverse of multiplication, which requires taking a number apart piece by piece, guessing at partial quotients, rather than just multiplying digit by digit and getting an answer directly. Inverses tend to take more work!