Answer
$$3$$
Work Step by Step
Given
\begin{aligned}
\:\int _0^{\frac{3\pi }{2}}\left|\sin\left(x\right)\right|dx\:
\end{aligned}
Since
\begin{aligned}
|\sin(x)|&= \begin{cases} \sin(x)&0\lt x\lt \pi\\
- \sin(x)&\pi\lt x\lt 3\pi/2
\end{cases}
\end{aligned}
Then
\begin{aligned}
\:\int _0^{\frac{3\pi }{2}}\left|\sin\left(x\right)\right|dx\: &= \int _0^{\pi }\sin \left(x\right)dx+\int _{\pi }^{\frac{3\pi }{2}}-\sin \left(x\right)dx\\
&= -\cos(x)\bigg|_0^{\pi}+ \cos(x)\bigg|_{\pi}^{3\pi/2}\\
&= 2+1=3
\end{aligned}
