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?
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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