Tích phân \(\int\limits^{\pi}_{\frac{\pi}{2}}x^2.\cos x\text{d}x\) bằng
\(2-2\pi-\dfrac{\pi^2}{4}\). \(2-2\pi+\dfrac{\pi^2}{4}\). \(1+2\pi-\dfrac{\pi^2}{4}\). \(1+2\pi+\dfrac{\pi^2}{4}\). Hướng dẫn giải:\(\int\limits^{\pi}_{\frac{\pi}{2}}x^2.\cos x\text{d}x=\int\limits^{\pi}_{\frac{\pi}{2}}x^2\left(\sin x\right)'\text{d}x=x^2\sin x|^{\pi}_{\frac{\pi}{2}}-\int\limits^{\pi}_{\frac{\pi}{2}}\sin x\left(x^2\right)'\text{d}x=-\left(\dfrac{\pi}{2}\right)^2-\int\limits^{\pi}_{\frac{\pi}{2}}\sin x.2x\text{d}x\)
\(=-\dfrac{\pi^2}{4}+\int\limits^{\pi}_{\frac{\pi}{2}}2x\left(\cos x\right)'\text{d}x=-\dfrac{\pi^2}{4}+2x\cos x|_{\frac{\pi}{2}}^{\pi}-\int\limits^{\pi}_{\frac{\pi}{2}}\cos x\left(2x\right)'\text{d}x\)
\(=-\dfrac{\pi^2}{4}-2\pi-\int\limits^{\pi}_{\frac{\pi}{2}}2\cos x\text{d}x=-\dfrac{\pi^2}{4}-2\pi-2\sin x|_{\frac{\pi}{2}}^{\pi}=-\dfrac{\pi^2}{4}-2\pi+2\).
Cách khác : dùng MTCT kiểm tra các đáp số).