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: \[\int \begin{pmatrix} x^2-x\end{pmatrix} \sqrt{ax+b}.dx\]
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.
Let \(u = \sqrt[n]{ax+b}\), that's \(u\) is the the entire radical expression.
In this case:
Using the fact that:
\[ax+b = u^n\]
Each of these two options, and how they work, are illustrated in the following tutorial, watch it now.
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.
Using the substitution of your choice, find each of the following integrals: