# Integration by Substitution for Radical Expressions

To integrate a product of functions, in which one of the functions is a composite radical function, like $$\sqrt{ax+b}$$ or $$\sqrt[3]{ax+b}$$, and the other is a polynomial function, like $$x^2+x$$ or $$x^3-x^2+1$$, we quickly see that U-Substitution won't work.

The method of substitution we learn in this section will allow us to integrate such proucts of functions.

Given an integral looking like: $\int \begin{pmatrix} x^2-x\end{pmatrix} \sqrt{ax+b}.dx$

or
$\int x^3 \sqrt[3]{ax+b}.dx$
or, more generally
$\int \begin{pmatrix}5x^2 +3 \end{pmatrix} \sqrt[n]{ax+b}.dx$
or ...
We have the choice between two substitutions:

### Option 1: let $$u = ax+b$$

Let $$u = ax+b$$, the linear function inside the radical expression.

In this case:

Using the fact that: $x = \frac{u-b}{a}$ We can rewrite the entire integral in terms of $$u$$ and integrate using the power rule for integration.

### Option 2: let $$u = \sqrt[n]{ax+b}$$

Let $$u = \sqrt[n]{ax+b}$$, that's $$u$$ is the the entire radical expression.

In this case:

Using the fact that: $ax+b = u^n$

and therefore:
$x = \frac{u^n -b}{a}$ We can rewrite the entire integral in terms of $$u$$ and integrate using the power rule for integration.

Each of these two options, and how they work, are illustrated in the following tutorial, watch it now.

## Tutorial

In the following tutorial we learn how to use the three equations to find the formula for the $$n^{\text{th}}$$ term of a quadratic sequence.

## Exercise 1

1. Find $$\int 3x \sqrt{x+4}.dx$$:

1. Using the substitution $$u = x +4$$

2. Using the substitution $$u = \sqrt{x +4}$$.

2. Find $$\int x^2 \sqrt[3]{x - 3}.dx$$:

1. using the substitution $$u = x-3$$

2. using the substitution $$u = \sqrt[3]{x - 3}$$.

## Exercise 2

Using the substitution of your choice, find each of the following integrals:

1. $$\int 3x \sqrt{x+4}.dx$$

2. $$\int 10x \sqrt[4]{2x+8}.dx$$

3. $$\int 5x \sqrt{2x+8}.dx$$

4. $$\int \begin{pmatrix}x^2+1 \end{pmatrix} \sqrt{x-5}.dx$$

5. $$\int 4x^2 \sqrt[3]{8x+3}.dx$$