The substitution rule: ∫ f(g(x))g'(x) dx = ∫ f(u) du
This rule is used to simplify certain integrals by replacing a complicated function inside the integral with a simpler function.
The rule states that if we have an integral of the form ∫ f(g(x))g'(x) dx, we can make a substitution u = g(x) to obtain the equivalent integral ∫ f(u) du. This is because, by the chain rule of differentiation, we have du/dx = g'(x), so dx = du / g'(x).
Therefore, substituting u = g(x) and dx = du / g'(x) in the original integral, we get:
∫ f(g(x))g'(x) dx = ∫ f(u) du / g'(x)
Multiplying both sides by g'(x), we obtain:
∫ f(g(x))g'(x) dx = ∫ f(u) du
which is the substitution rule.
This rule is especially useful when the function g(x) inside the integral is a polynomial, a trigonometric function, or an exponential function, as these are functions for which we know the antiderivatives. By making an appropriate substitution, we can simplify the integral and evaluate it more easily.