The method of u-substitution is an attempt to convert the given integral into a simpler integral by changing the variable. Here are the steps:
1. Decide what to let u equal.This is the hardest step.In deciding what to let u equal, two guidelines should be kept in mind:
a) The derivative of u (or the derivative of u off by a constant factor) should be elsewhere in the integral.
b) Your choice of u should simplify the integral.
2. Compute the derivative of u.
3. Solve for du by multiplying both sides by dx.
(Steps 2 and 3 can be combined into one step if you wish.)
4. If the expression on the right side of the equal sign in step 3 cannot be found in the original integral, fix this by multiplying both sides of the equation by an appropriate constant.
5. Substitute into the given integral, replacing everything that had the original variable (usually x) with an expression in terms of u.
6. Antidifferentiate the new integral using the methods of section 5.2.
7. Replace u with what it was originally equal to.

Let’s illustrate this method with an example.I will number the steps so you can see the correspondence with the steps above.You do not have to number your steps on a test.

1. u = x²

2. 

3. du = 2xdx

4. ½ du = xdx

5. 

6. -½cos u + C

7. -½cos(x²) + C

The proper choice of u is not that difficult since there are only a few choices.In the example above, the only possibilities were:

a. u = sin x²

b. u = x

c. u = x²

The first choice doesn’t work because the derivative of u, xcos(x²), cannot be found in the original integral.The second choice, u = x, should never be used since all it does is give you back the same integral in terms of u.