Integration By Substitution (Change Of Variable) - SS3 Mathematics Lesson Note

It is often better to transform the form of a function to make it easier to integrate.

Example: Evaluate \(\int_{}^{}{\sin{(2x + 3)}}\ dx\)

Solution:

\[\int_{}^{}{\sin{(2x + 3)}}\ dx\]

Let \(u = 2x + 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \therefore\frac{du}{dx} = 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \therefore dx = \frac{du}{2}\)

Substituting we have,

\(\int_{}^{}{\sin{(2x + 3)}}\ dx = \ \int_{}^{}{\sin{u\frac{du}{2} = \frac{1}{2}\int_{}^{}{\sin{u\ }du}}} = \frac{1}{2}\left( - \cos u \right) + C\), substitute \(u\)

\[= - \frac{1}{2}\cos(2x + 3) + C\]