# How do I evaluate int5y ln (9y) dy?

Jan 30, 2015

I would use integration by parts where you have:

$\int f \left(y\right) \cdot g \left(y\right) \mathrm{dy} = F \left(y\right) \cdot g \left(y\right) - \int F \left(y\right) \cdot g ' \left(y\right) \mathrm{dy}$

Where:

$F \left(y\right) = \int f \left(y\right) \mathrm{dy}$
$g ' \left(y\right)$ is the derivative of $g \left(y\right)$

In your case you can choose:

$f \left(y\right) = 5 y$
$g \left(y\right) = \ln \left(9 y\right)$

And so:

$\int 5 y \cdot \ln \left(9 y\right) \mathrm{dy} = 5 {y}^{2} / 2 \cdot \ln \left(9 y\right) - \int 5 {y}^{2} / 2 \cdot \frac{1}{9 y} \cdot 9 \mathrm{dy} =$
$= 5 {y}^{2} / 2 \cdot \ln \left(9 y\right) - \int 5 \frac{y}{2} \mathrm{dy} =$
$= 5 {y}^{2} / 2 \cdot \ln \left(9 y\right) - \frac{5}{4} \cdot {y}^{2} + c$