\(y'=-3.\dfrac{1}{3}.\cos^2x.\sin x+\dfrac{4}{\sin^2x}+\left(m+1\right)\sin x=\left(\sin^2-1\right)\sin x+\dfrac{4}{\sin^2x}+m.\sin x+\sin x\)
\(=\sin^3x+\dfrac{4}{\sin^2x}+m.\sin x\)
y đồng biến trên khoảng \(\left(0;\pi\right)\) \(\Leftrightarrow y'\ge0,\forall x\in\left(0;\pi\right)\)
\(\Leftrightarrow\sin^3x+\dfrac{4}{\sin^2x}+m.\sin x\ge0\Leftrightarrow\sin^2x+\dfrac{4}{\sin^3x}\ge-m\)
\(f\left(x\right)=\sin^2x+\dfrac{4}{\sin^3x}\Rightarrow f'\left(x\right)=2.\sin x.\cos x-\dfrac{12\cos x}{\sin^4x}=2\cos x.\left(\sin x-\dfrac{6}{\sin^4x}\right)\)
\(f'\left(x\right)=0\Rightarrow2\cos x\left(\sin x-\dfrac{6}{\sin^4x}\right)=0\)
\(\Rightarrow x=\dfrac{\pi}{2}\in\left[0;\pi\right]\)
\(\Rightarrow\sin^2x+\dfrac{4}{\sin^3x}\ge-m\Leftrightarrow-m\le min_{x\in\left(0;\pi\right)}f\left(x\right)\)
\(\Leftrightarrow m\ge-5\Rightarrow m\in\left\{-5;-4;-3;-2;-1\right\}\)
Có 5 giá trị m t/m
P/s: Mới học đạo hàm nên thử sức xí :v