Is there any theorem or axiom on which substitution is based?
Or substituting is it a mere art?
Thanks in advance.
Is there any theorem or axiom on which substitution is based?
Or substituting is it a mere art?
Thanks in advance.
I think this addresses your question:
It's the chain rule:
{d\over dx} \bigl(f(g(x)\bigr) =f'\bigl(g(x)\bigr)g'(x)
So, an antiderivative of the right hand side will give the function that's being differentiated on the left: \int f'\bigl(g(x)\bigr)g'(x) \,dx=\bigl(f(g(x)\bigr)+C.
So, when deciding upon the substitution, you want to recognize that the integrand is a product of functions where one term of the product is a composition of functions and the other term in the product is "almost" the derivative of the inner function of the composition (and that you can antidifferentiate the outer function). You then set $u$ equal to the inner function of the composition. \int f'(\underbrace{g(x)}_u) \underbrace{g'(x) \,dx}_{du}=\underbrace{(f(g(x))}_{\int u\,du}+C.
This, perhaps naive, rule of thumb will work in most cases. In other cases, the identification of the "proper" $u$ is an art...
Adding to the answer by David Mitra, more generally when learning calculus one should bear in mind that facts about indefinite integrals correspond to facts about derivatives. This is precisely because the operation of antidifferentiation is defined in terms of the operation of differentiation.
So, for example, $\frac{d}{dt}[kf+g] = k\frac{df}{dt} + \frac{dg}{dt}$ $\int kf(t)+g(t)\ dt = k\int f(t)\ dt + \int g(t)\ dt,$
and
$\frac{d}{dt}[fg] = \frac{df}{dt}g + f\frac{dg}{dt}$ f(t)g(t) = \int f(t)g'(t)\ dt + \int f'(t)g(t)\ dt (more commonly, integration by parts)
and, relevant to this thread,
[f(g(t))]' = f'(g(t))g'(t) \int f'(g(t))g'(t)\ dt = \int f'([g(t)])[g'(t)\ dt] = \int f'(u)\ du = f(u) = f(g(t)) + C, where I have carried out the substitution $u = g(t)$, du = g'(t)dt.
I apologize that this is slightly off topic, but I felt that this general philosophy would add to the asker's understanding of where all the various rules of integration come from.