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,

${\int }_{}^{}\sin (2x+3)dx={\int }_{}^{}\sin u\frac{du}{2}=\frac{1}{2}{\int }_{}^{}\sin udu=\frac{1}{2}(-\cos u)+C$, substitute $u$

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