Integration - the Australian Mathematical Sciences Institute

A guide for teachers – Years 11 and 12 • {19}Exercise 7Find the indefinite integrals∫1a d x b3 − 2x∫1(3 − 2x) 2 d x.The following theorem gives some useful rules for computing integrals.Theorem (Linearity of integration)aIf f and g are continuous functions, then∫ ∫∫(f )(x) ± g (x) d x = f (x) d x ± g (x) d x.bIf f is a continuous function and k is a real constant, then∫∫k f (x) d x = k f (x) d x.This theorem allows us to compute an antiderivative by treating a function term-by-termand factoring out constants, as the next example illustrates.ExampleFind∫(3x 3 − 4x 2 + 2) d x.SolutionUsing the above rules, we can rewrite the integral as follows:∫∫ ∫ ∫(3x 3 − 4x 2 + 2) d x = 3x 3 d x − 4x 2 d x + 2 d x∫= 3∫x 3 d x − 4∫x 2 d x + 21 d x= 3x44 − 4x3 + 2x + c.3Note. You might think that, since there are several indefinite integrals on the second lineof these equations, there should be several different constants in the answer. But sincethe sum of several constants is still a constant, we can just write a single constant.