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If \int\frac{\cos\theta}{5 + 7 \sin\theta -2\cos \theta}d\theta = A\log _e(B(\theta)) + C where C=constant of integration, thenB(\theta) =
Option: 1 \frac{2\sin\theta +1 }{5(\sin\theta +3)}
Option: 2 \frac{2\sin\theta +1 }{\sin\theta +3}
Option: 3 \frac{5(\sin\theta +3)}{2\sin\theta +1}
Option: 4 \frac{5(2\sin\theta +1)}{\sin\theta +3}

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best_answer

\\\int \frac{\cos \theta}{5+7 \sin \theta-2\left(1-\sin ^{2} \theta\right)} d \theta \\ \int \frac{\cos \theta}{2 \sin ^{2} \theta+7 \sin \theta+3} d \theta \\ \int\frac{\cos \theta}{(2 \sin \theta+3)(\sin \theta+1)} d \theta

\\\text { Let } \sin \theta=t\qquad \Rightarrow\qquad \cos \theta d \theta=d t \\\\ \int \frac{d t}{(2 t+1)}(t+3)

\left[\frac{2}{5} \int \frac{d t}{2 t+1}-\frac{1}{5} \int \frac{d t}{2 t+3}\right]

\\\frac{2}{5} \ln (2 \sin \theta+1)-\frac{1}{5} \ln (\sin \theta+3)+C \\ \\\frac{1}{5} \ln \left(\frac{(2 \sin \theta+1)}{\sin \theta+3}\right)+C

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himanshu.meshram

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