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 1Edit

**Integrand contains**

Use the substitution .

**Example**

Find .

### Type 2Edit

**Integral is of the form**

Write as .

**Example**

Find .

### Type 3Edit

**Integrand contains** , or

This was discussed in "trigonometric substitutions above". Here is a summary:

- For , use .
- For , use .
- For , use .

### Type 4Edit

**Integral is of the form**

Use the substitution .

**Example**

Find .

### Type 5Edit

**Other 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