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Câu hỏi: Tìm nguyên hàm của hàm số $f\left( x \right)=\dfrac{\cos x}{\sin x+2}$.
A. $\dfrac{1}{\sin x+2}+C.$
B. $\ln \left( \sin x+2 \right)+C.$
C. $-\dfrac{1}{\sin x+2}+C.$
D. $-\ln \left( \sin x+2 \right)+C.$
A. $\dfrac{1}{\sin x+2}+C.$
B. $\ln \left( \sin x+2 \right)+C.$
C. $-\dfrac{1}{\sin x+2}+C.$
D. $-\ln \left( \sin x+2 \right)+C.$
Ta có: $\int{\dfrac{\cos x}{\sin x+2}dx=\int{\dfrac{1}{\sin x+2}d\left( \sin x \right)=\int{\dfrac{1}{\sin x+2}d\left( \sin x+2 \right)}=\ln \left| \sin x+2 \right|+C}}$.
Mà $\sin x+2\ge -1+2>0\Rightarrow \int{\dfrac{\cos x}{\sin x+2}dx=\ln \left( \sin x+2 \right)+C}$.
Mà $\sin x+2\ge -1+2>0\Rightarrow \int{\dfrac{\cos x}{\sin x+2}dx=\ln \left( \sin x+2 \right)+C}$.
Đáp án B.