Integration of a composite function

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Homework Help Overview

The discussion revolves around the integration of composite functions, specifically the integral of the form ∫f(g(x)) dx. Participants are exploring the general principles of integration in this context.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the potential use of substitution as a technique for integrating composite functions, noting that its applicability may depend on the specific forms of f and g. There is a question about the existence of a general method for integration similar to that for differentiation.

Discussion Status

The conversation is ongoing, with some participants suggesting substitution as a viable approach while others express uncertainty about the existence of a universal method for integration of composite functions. The mention of the relationship between the derivative of the inside function and the outside function indicates a productive line of reasoning.

Contextual Notes

Participants are operating under constraints due to a lack of access to integration resources, which may influence their understanding and exploration of the topic.

Nisheeth
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Homework Statement


The question I have is a more general one, but one I can't find an anser to since I don't have any access to a book on integration at the moment.

How do we Integrate a composite function.

∫f(g(x)) dx

Homework Equations



The Attempt at a Solution

 
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A technique that might work is substitution. But it depends on the specific values of f and g, really.
 
Ok, so that means that there is not general method like for integration, as there is for differentiation?
Nonetheless thanks!
 
Nisheeth said:
Ok, so that means that there is not general method like for integration, as there is for differentiation?
Nonetheless thanks!

If you have ∫f(g(x))g'(x)dx, then this is equal to the simpler ∫f(u)du. This is the analogue to the chain rule for integrals. I say it in words as, the derivative of the inside must appear outside.
 

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