partial fraction decomposition in integral I have a question that asks to calculate the following integral:
$$ \int_0^\infty {\frac{w\cdot \sin w}{4a^2+w^2}dw} $$
In the official solution they used partial fraction decomposition in order to later use Plancherel's identity:
$$ \frac{w\cdot \sin w}{4a^2+w^2} =  $$
$$ \frac{\sin w}{w}\cdot\frac{w^2}{w^2+4a^2} = $$
$$ \frac{\sin w}{w}(1-\frac{4a^2}{w^2+4a^2}) = $$
$$ \frac{\sin w}{w} - \frac{\sin w}{w} \cdot \frac{4a^2}{w^2+4a^2}  $$
And then used Plancherel's identity.
But I didn't understand how to expand to partial fraction and in particular I didn't understand this equation:
$$ \frac{\sin w}{w}\cdot\frac{w^2}{w^2+4a^2} = \frac{\sin w}{w}(1-\frac{4a^2}{w^2+4a^2}) $$
Can you please explain how to expand the integrand into partial fraction?

$$1=\frac{w^2}{w^2+4a^2}+\frac{4a^2}{w^2+4a^2}$$
