It is possible to transform a difficult integral to an easier integral in a different variable using a substitution. By using substitutions, we can show that:
Find the integral of:
(a) sin x cos²x
x³ + 1
(a) Using the first of the two above formulae above, imagine f(x) = cos x. Therefore [f(x)]² = cos²x and f '(x) = sin x. Therefore, since n = 2, the answer is simply (cos³x)/ 3 + c
(b) Since the top is the differential of the bottom, we can use the second of the two formulae above to get the answer of ln(x³ + 1) + c.
Using a Substitution
Sometimes you will be told to integrate a function by using a substitution.