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Method of Substitution for Integrating the Product of a Linear Radical and a Polynomial

Say we're asked to find the following integrals: \[\int \begin{pmatrix} x^2 - 1 \end{pmatrix} \sqrt{x+2}.dx, \quad \int 3x \sqrt[4]{4x - 5} dx\] where, in each case, the integrand is a product of two functions:

  • a polynomial function, \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots +a_1x + a_0\)
  • a radical function whose radicand is a linear, \(g(x) = \sqrt[n]{ax+b}\)
then we use the method of substitution for integration, in other words we make a change of variable, in either one of two ways:

Method

To find integrals, like those shown above, we use the method of substitution by making a change of variable, \(u\), using either of the following two options:

Option 1: use the subsitution \(u=ax+b\)

Option 2: use the subsitution \(u=\sqrt[n]{ax+b}\)

Say we have to find the integral \(\int \begin{pmatrix}x^2 + 1 \end{pmatrix} \sqrt{2x-1}. dx\), then we could use either of the following two substitutions:

  • let \(u = 2x - 1\), or
  • let \(u = \sqrt{2x - 1}\)
both options will result in the same final answer. Watch the following tutorials to see how it all works.

Option 1: \(u = 2x - 1\)

In the following tutorial we show how to find the integral \(\int \begin{pmatrix}x^2 + 1 \end{pmatrix} \sqrt{2x-1}. dx\), using the change of variable \(u = 2x - 1\).

Option 2: \(u = \sqrt{2x-1}\)

In the following tutorial we show how to find the integral \(\int \begin{pmatrix}x^2 + 1 \end{pmatrix} \sqrt{2x-1}. dx\), using the change of variable \(u = \sqrt{2x-1}\).

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 \(\int x^2 \sqrt[3]{2x - 3} .dx\):
    1. using the substitution \(u = 2x - 3\)
    2. using the substitution \(u = \sqrt[3]{2x-3}\)

Note: this exercise can be downloaded as a worksheet to practice with: Worksheet 1

Exercise 2

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

  1. \(\int 5x \sqrt{x - 2}.dx\)
  2. \(\int 10x \sqrt[4]{2x+8}.dx\)
  3. \(\int \begin{pmatrix} 3x + 1 \end{pmatrix} \sqrt{ \frac{x+3}{2}}.dx\)
  4. \(\int \begin{pmatrix}x^2+1 \end{pmatrix} \sqrt{x-5}.dx\)
  5. \(\int \begin{pmatrix} x^2 + 2x \end{pmatrix} \sqrt[3]{\frac{x-1}{2}}.dx\)

Note: this exercise can be downloaded as a worksheet to practice with: Worksheet 2

Solution Without Working


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