Is there a good explanation of u substitution (or any other integral stuff that makes use of the denominator) using non-Liebnitz notation? Is it just that ∂1 is actually ∂/∂1, and thus you can begin using ∂1 the way you would have used ∂x?
You have ∫ f(g(x)) g'(x) dx. [So it might be something like ∫ [cos(x^2)] (2 x) dx, where f(x) = cos x and g(x) = x^2.] You do the u-substitution u = g(x), so u'(x) = g'(x):
∫ f(g(x)) g'(x) dx = ∫ f(u) u'(x) dx
Let F(u) = ∫ f(u) du: F is the antiderivative of f. By definition of the antiderivative, F'(u) = f(u). So by the Chain Rule,
d/dx F(u) = F'(u) u'(x) = f(u) u'(x).
∫ f(g(x)) g'(x) dx = ∫ f(u) u'(x) dx = F(u) = ∫ f(u) du.