An effective method of integration is integration by algebraic substitution. In this type of problem ^{dy}⁄_{dx} can be treated like a fraction.
The key to recognising this type of integration is that one part of the expression to be integrated is often the derivative of another part. Make sure that x is totally removed from the substituted integral.
The best way to illustrate integration by substitution is by six examples of some of the different methods.
Example 1

Example 2

Example 3

Find ∫ (3x − 7)^{5} dx  Find ∫ x(x − 3)^{4} dx  Find ∫ x √(x^{2} + 9) dx 
Example 4

Example 5

Example 6

Find ∫ 8e^{x4}. x^{3 }dx  Find  Find 
"Although each example has a similar substitution, each has its own particular features and as always, practice makes perfect!"
