Question

$\int \frac{3\sin x+2\cos x}{3\cos x+2\sin x}dx=$

• A. $\frac{12}{13}x-\frac{5}{13}\log |3\cos x+2\sin x|+c$
• B. $\frac{12}{13}x+\frac{5}{13}\log |3\cos x+2\sin x|+c$
• C. $\frac{5}{13}x-\frac{12}{13}\log |3\cos x+2\sin x|+c$
• D. $\frac{5}{13}x+\frac{12}{13}\log |3\cos x+2\sin x|+c$

Answer 1

The correct option is A $\frac{12}{13}x-\frac{5}{13}\log |3\cos x+2\sin x|+c$

$N\left(x\right)=3\sin \left(x\right)+2\cos x$

$D\left(x\right)=3\cos \left(x\right)+2\sin \left(x\right)$

$N\left(x\right)=\lambda D^1\left(x\right)+\mu \left(D\right(x\left)\right)$

$3\sin x+2\cos x=\lambda (-3\sin x+2\cos x)+\mu (3\cos x+2\sin x)$

$\Rightarrow \begin{array}{c}-3\lambda +2\mu =3\\ 2\lambda +3\mu =2\end{array}\}\begin{array}{c}13\mu =12\Rightarrow \mu =\frac{12}{13}\\ 13\lambda =-5\Rightarrow \lambda =\frac{-5}{13}\end{array}$

$N\left(x\right)=\frac{-5}{13}{D}^1\left(x\right)+\frac{12}{13}D\left(x\right)$

$\int \frac{N\left(x\right)}{D\left(x\right)}dx=\frac{-5}{13}\int \frac{D^1\left(x\right)}{D\left(x\right)}+\frac{12}{13}\int \frac{D\left(x\right)}{D\left(x\right)}\cdot dx$

$=\frac{-5}{13}\log \left|D\right(x\left)\right|+\frac{12}{13}x+c$

$=\frac{12}{13}x\frac{-5}{13}\log |3\cos x+2\sin x|+c$

$\int \frac{3\sin x+2\cos x}{3\cos x+2\sin x}dx=\frac{12}{13}x\frac{-5}{13}\log |3\cos x+2\sin x|+c$