Calculus/Integration techniques/Irrational Functions
Integration of irrational functions is more difficult than rational functions, and many cannot be done. However, there are some particular types that can be reduced to rational forms by suitable substitutions.
Type 1
editIntegrand contains
Use the substitution .
- Example
Find .
Find .
Type 2
editIntegral is of the form
Write as .
- Example
Find .
Type 3
editIntegrand contains , or
This was discussed in "trigonometric substitutions above". Here is a summary:
- For , use .
- For , use .
- For , use .
Type 4
editIntegral is of the form
Use the substitution .
- Example
Find .
Type 5
editOther rational expressions with the irrational function
- If , we can use .
- If , we can use .
- If can be factored as , we can use .
- If and can be factored as , we can use