# Integration by substitution

Source: https://projectalevel.co.uk/maths/calculus/integration_by_substitution

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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:

![](https://content.projectalevel.co.uk/img/maths/subs1.gif)

Example:
Find the integral of:
(a) sin x cos²x
(b) 3x²
 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.

![](https://content.projectalevel.co.uk/img/maths/subs2.gif)