\input amstex

\def\dd{\,\hbox{d}}

$$\aligned
\int_0^{5\pi/6}\sin x\dd x&=\int_0^{\pi/2}\sin x\dd x+\int_{\pi/2}^{5\pi/6}\sin x\dd x=\int_0^1\frac{t\dd t}{\sqrt{1-t^2}}-\int_1^{1/2}\frac{t\dd t}{\sqrt{1-t^2}}=\cr
&=-\sqrt{1-t^2}\Big|_0^1+\sqrt{1-t^2}\Big|_1^{1/2}=1+\frac{\sqrt{3}}{2}.
\endaligned$$
\bye