110. A random variable $X$, all possible values of which belong to the interval ( $0 ; \pi / 3$ ), is given in this interval by the probability density function $f(x)=C \cdot \sin 3 x$. Find the coefficient $C$.
Solution. If the function $f(x)$ represents the probability density function of a continuous random variable $X$, defined in the interval $(a, b)$, then the condition $\int_{a}^{b} f(x) d x=1$ is satisfied.

To determine the unknown coefficient $C$ in the expression of the function $f(x)=C \cdot \sin 3 x$ so that this condition is met for the given random variable $X$, we will find

$$
\int_{0}^{\pi / 3} f(x) d x=\int_{0}^{\pi / 3} C \cdot \sin 3 x d x=-\frac{C}{3}(\cos \pi-\cos 0)=-\frac{C}{3}(-1-1)=\frac{2 C}{3}
$$

and equate this result to one: $\frac{2 C}{3}=1$. From the last equation, we get $C=3 / 2$.