Biết m là số thực thỏa mãn

D

- Hướng dẫn giải

Đáp án D.

Ta có:  $\underset{0}{\overset{\frac{\pi }{2}}{\int }}x\left(\text{cosx+2m}\right)dx=\underset{0}{\overset{\frac{\pi }{2}}{\int }}x\cos xdx+\underset{0}{\overset{\frac{\pi }{2}}{\int }}2mxdx=I_1+I_2$

Ta có:  $I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}x\cos xdx=\underset{0}{\overset{\frac{\pi }{2}}{\int }}xd\sin x=x\sin x\left|\begin{array}{l}{}^{\frac{\pi }{2}}\\ {}_0\end{array}\right.-\underset{0}{\overset{\frac{\pi }{2}}{\int }}\sin \text{x}dx=\frac{\pi }{2}+c\text{osx}\left|\begin{array}{l}{}^{\frac{\pi }{2}}\\ {}_0\end{array}\right.=\frac{\pi }{2}-1$

 $I_2=m{x}^2\left|\begin{array}{l}{}^{\frac{\pi }{2}}\\ 0\end{array}\right.=m\frac{{\pi }^2}{4}\Rightarrow I_1+I_2=\frac{m{\pi }^2}{4}+\frac{\pi }{2}-1\Rightarrow m=8.$