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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.

Method - Radical Expressions

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 \(u = ax+b\), 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 \(u = \sqrt[n]{ax+b}\), the entire raidical 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 \(u = \sqrt[3]{x - 3}\).

Answers Without Working

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

Solution Without Working

Solution With Working