Google
Câu hỏi: Họ nguyên hàm của hàm số $f\left( x \right)=\dfrac{4\sin x+3\cos x}{sinx+2\cos x}$ có dạng $F\left( x \right)=ax+b\ln \left| sinx+2\cos x \right|$. Tính $P=\dfrac{-b}{a}$
A. $\dfrac{-1}{2}$
B. $\dfrac{1}{2}$
C. 2
D. -2
A. $\dfrac{-1}{2}$
B. $\dfrac{1}{2}$
C. 2
D. -2
Có $4\sin x+3\cos x=A\left( \sin x+2\cos x \right)+B\left( \cos x-2\sin x \right)$
$\begin{aligned}
& \Rightarrow \left\{ \begin{aligned}
& A-2B=4 \\
& 2A+B=3 \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& A=2 \\
& B=-1 \\
\end{aligned} \right. \\
& \Rightarrow 4\sin x+3\cos x=2\left( \sin x+2\cos x \right)-\left( \cos x-2\sin x \right) \\
\end{aligned}$
Suy ra $\int{f\left( x \right)dx=\int{\left( 2-\dfrac{\cos x-2\sin x}{\sin x+2\cos x} \right)dx=2x-\ln \left| \sin x+2\cos x \right|}}$
$\Rightarrow a=2;b=-1\Rightarrow P=-\dfrac{b}{a}=\dfrac{1}{2}$
$\begin{aligned}
& \Rightarrow \left\{ \begin{aligned}
& A-2B=4 \\
& 2A+B=3 \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& A=2 \\
& B=-1 \\
\end{aligned} \right. \\
& \Rightarrow 4\sin x+3\cos x=2\left( \sin x+2\cos x \right)-\left( \cos x-2\sin x \right) \\
\end{aligned}$
Suy ra $\int{f\left( x \right)dx=\int{\left( 2-\dfrac{\cos x-2\sin x}{\sin x+2\cos x} \right)dx=2x-\ln \left| \sin x+2\cos x \right|}}$
$\Rightarrow a=2;b=-1\Rightarrow P=-\dfrac{b}{a}=\dfrac{1}{2}$
Đáp án B.