Mathematics - Indefinite Integration Question with Solution | TestHub

$I=\int \frac{\cos \theta }{2{\sin }^2\theta +7\sin \theta +3}d\theta$

$\sin \theta =t \Rightarrow \cos \theta d\theta =dt$

$=\frac{1}{2}\int \frac{1}{t^2+\frac{7}{2}t+\frac{3}{2}}dt=\frac{1}{2}\int \frac{1}{{\left(t+\frac{7}{4}\right)}^2-{\left(\frac{5}{4}\right)}^2}dt=\frac{1}{5}\ln \left|\frac{2t+1}{t+3}\right|+c=\frac{1}{5}\ln \left|\frac{2\sin \theta +1}{\sin \theta +3}\right|+c$

so $A=\frac{1}{5}$

$B\left(\theta \right)=\frac{5(2\sin \theta +1)}{\sin \theta +3}$