Can I Use Functions Within Functions for Integration?

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Functions can be nested within each other for integration, allowing for expressions like f(P) where P is a variable of integration. The example of integrating \int \sin(x^2) d(x^2) demonstrates that this approach is valid, yielding -\cos(x^2). However, integrating \int \sin(x^2) dx does not yield the same result, highlighting the importance of the variable of integration. This discussion relates closely to the method of "u-substitution," which facilitates the integration of composite functions. Understanding these concepts is crucial for accurate integration in calculus.
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Hello,

I am curious, suppose I have a function of x, f(x). Suppose I also have another function P(x). Does this mean I am allowed to have f(P) and I can do standard methods of integration and such on it?
 
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If P is the variable that you are integrating. For example, \int \sin(x^2) d(x^2) can be computed the way it seems you want to compute it. That is, \int \sin(x^2) d(x^2) = - \cos(x^2). However, \int \sin(x^2) dx \neq - \cos(x^2). The question you are asking has a lot to do with the integration method known as "u-substitution." Do you know what that is?
 
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