If $\int_0^{2\pi} |x \sin x| \, dx = k\pi$, then the value of $k$ is equal to _____.

Over the interval $[0, 2\pi]$, $x \ge 0$, so $|x \sin x| = x |\sin x|$.

We split the integral at $x = \pi$ where $\sin x$ changes sign:

• For $x \in [0, \pi]$, $\sin x \ge 0 \implies |x \sin x| = x \sin x$

• For $x \in [\pi, 2\pi]$, $\sin x \le 0 \implies |x \sin x| = -x \sin x$

Using integration by parts, $\int x \sin x \, dx = -x \cos x + \sin x$:

1. First integral:

2. Second integral:

Combining the two:

Since this equals $k\pi$, we have $k = 4$.