A continuous random variable X has the probability density function $f$ given by

.

Find P(0 ≤ X ≤ 3).

attempting integration by parts, eg

$u=\frac{\pi x}{36}\text{,}\text{d}u=\frac{\pi }{36}\text{d}x\text{,}\text{d}v=\text{sin}\left(\frac{\pi x}{6}\right)\text{d}x\text{,}v=-\frac{6}{\pi }\text{cos}\left(\frac{\pi x}{6}\right)$               (M1)

P(0 ≤ X ≤ 3) $=\frac{\pi }{36}\left({\left[-\frac{6x}{\pi }\text{cos}\left(\frac{\pi x}{6}\right)\right]}_0^3+\frac{6}{\pi }\underset{0}{\overset{3}{\int }}\text{cos}\left(\frac{\pi x}{6}\right)\text{d}x\right)$ (or equivalent)      A1A1

Note: Award A1 for a correct $uv$ and A1 for a correct $\int v\text{d}u$.

attempting to substitute limits       M1

$\frac{\pi }{36}{\left[-\frac{6x}{\pi }\text{cos}\left(\frac{\pi x}{6}\right)\right]}_0^3=0$       (A1)

so P(0 ≤ X ≤ 3) $=\frac{1}{\pi }{\left[\text{sin}\left(\frac{\pi x}{6}\right)\right]}_0^3$ (or equivalent)       A1

$=\frac{1}{\pi }$      A1

[7 marks]