However, even with integration by parts taken together the two substitutions must completely replace the original variable, with no leftovers allowed. In that case, you do two u-substitutions (but you call them by different variables). It isn't really an exception, but you can sort of have leftover bits when you do integration by parts. ∫ cos (x²) dx is nightmarishly difficult (getting into something called Fresnel integrals). ∫ (x)∙cos(x²) dx is very easy to integrate but the very similar looking With integration, being close to a standard form is not good enough: you must have an exact match. So, the answer is, no, you cannot do u-substitution that way. And remember du is the derivative of whatever you called u, it is NOT just some notation.

Where f(u) du is something you know how to integrate.

The purpose of u substitution is to wind up with ∫ f(u) du A valid substitution, generally speaking, requires that ALL references to the original variable be replaced ESPECIALLY including its dx (or whatever the variable is). Sorry, that is ordinarily not a valid means of substitution.

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