Use substitution to find ʃ x(x^2 – 4)^2 dx

Answer 1

When you find an integral by substitution, you are essentially doing the Chain Rule (for differentiation) in reverse.

Recall the Chain Rule says that if h(x) = f(g(x)), then h'(x) = f'(g(x))*g'(x).

So if you have an integrand that contains both a function g(x) and its derivative g'(x), it can sometimes be useful to try the substitution u = g(x), with du = g'(x)dx.

For example, if you were trying to find ∫sin³(x)cos(x)dx, you might notice that both sin(x) and its derivative cos(x) appear in the integrand. So let u = sin(x), then du = cos(x)dx and the integral becomes ∫u³du, which is simply (1/4)u⁴ + C = (1/4)sin⁴(x) + C.

So do you see any functions in your integrand that also have their derivative present as well (to within a constant factor, which can always be pulled outside the integral)?