求 Sin 2021X 比上 sin x 在 0~2 分之派上的定积分。

Interpreting the query

The user seems to want to compute the definite integral of $\frac{\sin(2021x)}{\sin(x)}$ from 0 to $\frac{2\pi}{2}$, but there's some uncertainty in the translation. "求" means "find," and "sin 2021X 比上 sin x" translates as $\frac{\sin(2021x)}{\sin(x)}$. The “0~2 分之派” likely means "from 0 to $\pi$," as "2分之π" is a common notation for $\frac{\pi}{2}$. So, the calculation should be from 0 to $\pi$.

Clarifying the integral range

The user’s question involves $\frac{\sin(2021x)}{\sin(x)}$, with a limit of integration from 0 to possibly $\frac{\pi}{2}$ (since "2分之派" translates to $\frac{\pi}{2}$) or maybe $2\pi$. There’s a known formula that $\int_0^\pi \frac{\sin(mx)}{\sin(x)} dx = \pi$ when $m$ is odd, and the result is zero if $m$ is even. Based on this, if the interval is from 0 to $\pi$, the result would be $\pi$. If it's from 0 to $2\pi$, the result would be $2\pi$.

Evaluating the integral limits

The user is trying to evaluate an integral of $\frac{\sin(2021x)}{\sin x}$, and there’s some confusion about the interval. The phrase "2分之派" could mean $\frac{\pi}{2}$, so the interval might be from 0 to $\frac{\pi}{2}$. There's an identity for $\int_0^\pi \frac{\sin((2n+1)x)}{\sin(x)} dx = \pi$, and adjusting this for our specific case, it seems the integral from 0 to $\frac{\pi}{2}$ will give $\frac{\pi}{2}$.

Reasoned for 19 seconds