In indefinite integral section, various formulas have been introduced to calculate the integral of a function. However, it is not possible to have a formula for every possible situation. Thus, basic formulas like

need to be generalized so they can be used in a variety of cases. This rule is called the substitution rule.

The substitution rule states:

If u = g(x) is a differentiable function in an interval R and f is continuous on R, then

Original Function f(x)

Substitution Function for f(x)

This rule can be better understand by calculating      .

Let u = 5x² + 7. The derivative of u with respect to x is

     du/dx = 10x

Rearrange the above equation gives

     du = 10xdx

Thus

In order to double check the answer, the derivative of function can be taken.

The result confirms the correction of .

Substitution rule can also be used for definite integrals. The substitution rule for definite integrals states:

If g'(x) is continuous on interval R and f(x) is continuous on the range of g(x), then

This rule can be better understand by calculating .

Let u = 5x² + 7. The derivative of u with respect to x is

     du/dx = 10x

Rearrange the above equation gives

     du = 10xdx

The new limits of integration can be calculated. When x = 0, u = 7 and when x = 1, u = 12. Therefore,

It is obviously that the substitution rule provides an easy way to calculate the integral of a function.