In describing methods of integration of irrational functions (containing root functions) we will restrict ourselves to classes of great practical importance.

Integration of functions containing a root of a linear expression

An integral

where R is a rational function with two arguments, can be reduced to an integral of a rational function by application of the substitution

Example

Evaluate the integral

Integration of functions containing a square root of an arbitrary polynomial of the second degree

It can be shown that an integral of the form

where R is a rational function with two arguments, can be expressed in terms of elementary functions. It can be effectively calculated by reducing to one of the following integrals

The first one is the basic integral

The second one, assuming x ² + k > 0, will be reduced to the integral of a rational function by using Euler's substitution

we have

and

Hence

And finally

Example

Evaluate the integral

Transforming the trinomial into the canonical form

and substituting t = x - 3 we get

If a < 0, then the integral

can be reduced in a similar way to the integral of the first type.

In order to calculate the integral

we represent it in the form

The first integral can be found by integrating by parts

and the second one is already known. Thus we obtain

and finally

Likewise we could find the integral

However, applying the substitution x = sint, we can get the result much faster:

Exercise 8.8

Using the substitution x = sint evaluate the integral