The general power rule allows one to integrate functions where the function is raised to a power by treating it as the product of that function and its derivative, raised to the same power minus one, divided by that new power. For example, to integrate x^3 one would write the integral as x^2 * dx / 2. However, the general power rule does not always work, so an alternative is to use u-substitution to substitute the function for a variable u and the derivative of the function for du before integrating.

1. 5.2 General Power Rule and Integration:Goal: - to integrate functionsGeneral Power RuleIf u is a differentiable function of x then To use this you must identify what is the function and what is the derivative of the function.

2. i.e.Function part derivative part

3. Examples:u = du = u = du =

4. u = du =

7. Sometimes the general power rule does not work – what else can we try?

8. Substitution

9. The general power rule takes the form: We identify u and du then integrate.