## Example 7.2.4(a): Applying the Substitution Rule

The expression that makes the integral difficult is*4x + 3*so that we make the change of variable

Thenu = 4x + 3

which at first glance does not appear in the original integral. But:du/dx = 4, ordu = 4 dx

There is no "incorrect" substitution, you can make any change of variables that you want. There(4x + 3)^{2}dx = u^{2}1/4 4 dx =

= 1/4 u^{2}4 dx = 1/4 u^{2}du =

= 1/4 1/3 [(4b + 3)^{3}- (4a + 3)^{3}]

*are*, however, substitutions that work (such as above) and those that do not work (which are not incorrect but not successful). For example, we could decide to make the following substitution in the above example:

Thenu = (4x + 3)^{2}

The new integral looks likedu/dx = 2 (4x + 3) 4ordu = 8 (4x + 3) dx

*u dx*, but it is not (easily) possible to remove the

*dx*. Therefore, this substitution, while correct, was not successful.