Recognisable Forms and Substitution

The H2 syllabus emphasises integrals in which an inner function and its derivative appear together. These forms are consequences of the chain rule.

Core recognisable forms

Change of variables

Let \(u=\mathrm{g}(x)\), where \(\mathrm{g}\) is differentiable and its range is an interval \(I\). If \(\mathrm{f}\) is continuous on \(I\), then

\[\int \mathrm{f}(\mathrm{g}(x))\mathrm{g}'(x)\,\mathrm{d}x=\int \mathrm{f}(u)\,\mathrm{d}u\]

This method is also called substitution or the reverse chain rule.

Substitution sequence

1. Choose \(u\) so that the remaining factors can be written using \(u\) and \(\mathrm{d}u\).
2. Rewrite the whole integrand; do not mix \(x\) and \(u\) in the transformed integral.
3. After integrating, substitute back for \(u\) and include \(C\).

Illustrative substitutions

Cancelling Method (u-substitute method)

Simple case: \(\displaystyle\int x(x^2+1) \, \mathrm{d}x = \displaystyle\int x^3 + x \, \mathrm{d}x = \frac{x^4}{4} + \frac{x^2}{2} + c\)

Complex case: \(\displaystyle\int x(x^2+1)^{10} \, \mathrm{d}x\)
Let \(u = x^2 + 1\), \(\displaystyle\frac{\mathrm{d}u}{\mathrm{d}x} = 2x\), \(\mathrm{d}x = \frac{\mathrm{d}u}{2x}\)
\(= \displaystyle\int x \cdot u^{10} \cdot \frac{\mathrm{d}u}{2x} = \frac{1}{2}\displaystyle\int u^{10}\,\mathrm{d}u\) [see MF26]
\(= \displaystyle\frac{1}{22}(x^2+1)^{11} + c\)

More Examples:

• \(\displaystyle\int \sec^2 x (2\tan x + 3)^5 \, \mathrm{d}x\): let \(u = 2\tan x + 3\)
\(= \displaystyle\frac{1}{12}(2\tan x + 3)^6 + c\)

• \(\displaystyle\int \frac{x^2}{x^3-1} \, \mathrm{d}x\): let \(u = x^3 - 1\)
\(= \displaystyle\frac{1}{3}\displaystyle\int \frac{1}{u}\,\mathrm{d}u = \frac{1}{3}\ln|x^3-1| + c\)

• \(\displaystyle\int \frac{e^x}{1+e^x} \, \mathrm{d}x\): let \(u = 1 + e^x\)
\(= \ln|1+e^x| + c\)