Giá trị của \(\sin\left(\int\limits_0^{\pi}\pi x\cos x\text{d}x\right)\) là
\(1\).\(0\).\(\pi\).\(\dfrac{\sqrt{3}}{2}\).Hướng dẫn giải:Ta có:
\(\int\limits_0^{\pi}\pi x\cos x\text{d}x=\pi\int\limits^{\pi}_0x\cos x\text{d}x\)
Đặt \(\begin{cases}u=x\\v'=\cos x\end{cases}\) \(\Rightarrow\begin{cases}u'=1\\v=\sin x\end{cases}\)
\(\pi\int\limits_0^{\pi}x\cos x\text{d}x=\pi\left(x.\sin x|^{\pi}_0-\int\limits^{\pi}_0\sin x\text{d}x\right)\)
\(=\pi\left(x\sin x|^{\pi}_0+\cos x|^{\pi}_0\right)\)
\(=-2\pi\)
Suy ra:
\(\sin\left(\int\limits_0^{\pi}\pi x\cos x\text{d}x\right)=\sin\left(-2\pi\right)=0\) . Chọn \(\sin\left(\int\limits^{\pi}_0\pi x\cos x\text{d}x\right)=0\).