Integration Methods

 

 Integration by Parts

 Integration by Substitution

Integration by Partial Fractions

 

Integration by Parts

 

This method is based on the rule for differentiating the product of two functions. We know that this rule is:

Integrating both parts:

And finaly:

This method is used to eliminate one of the functions inside the integral. So we actually change the form of the function to be integrated.

 

Example 1:

 

 Example 2:

 

Example 3:

This is a more complex example because we have an intermediate production of the original function.

 

Integration by Substitution

Integration by substitution allows the change of the basic variable of an integrand to another variable, in order to transform the integrand to a known form. The relationship between the old and the new variables must be specified.

 When the integration is completed, we substitute back the old variable through the inverse relation between the new and the old variable. 

 

Example 1:

Setting:

Substituting:

Substituting back:

 

Example 2:

Setting:

Substituting: 

Substituting back:

 

Example 3:

Setting:

Substituting:

Substituting back:

 

Integration by Partial Fractions

When the integrand is an algebraic fraction, using the partial fraction techniques, the integrant can be broken down into simpler and smaller fractions and then the integration can be easily calculated. To learn more about partial fractions you can see the appropriate chapter.

 

Example 1:

 

Example 2: