If $\int \frac{\cos \theta}{5+7 \sin \theta-2 \cos ^{2} \theta} d \theta=A \log _{e}|B(\theta)|+C$
where $C$ is a constant of integration, then $\frac{B(\theta)}{A}$
can be :
Correct Option: , 4
$\int \frac{\cos \theta d \theta}{5+7 \sin \theta-2 \cos ^{2} \theta}$
$\int \frac{\cos \theta d \theta}{3+7 \sin \theta+2 \sin ^{2} \theta} \quad \begin{aligned}&\sin \theta=t \\&\cos \theta d \theta=d t\end{aligned}$
$\int \frac{d t}{2 t^{2}+7 t+3}=\int \frac{d t}{(2 t+1)(t+3)}$
$=\frac{1}{5} \int\left(\frac{2}{2 t+1}-\frac{1}{t+3}\right) d t$
$=\frac{1}{5} \ln \left|\frac{2 \mathrm{t}+1}{\mathrm{t}+3}\right|+\mathrm{C}$
$=\frac{1}{5} \ln \left|\frac{2 \sin \theta+1}{\sin \theta+3}\right|+\mathrm{C}$
$\mathrm{A}=\frac{1}{5}$ and $\mathrm{B}(\theta)=\frac{2 \sin \theta+1}{\sin \theta+3}$
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