Question

Write a value of $\int \frac{\cos x}{3+2\sin x}dx$.

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int \frac{\cos x}{3+2\sin x}dx \\ \mathrm{Let}3+2\sin x=t \\ \Rightarrow 2\cos xdx=dt \\ \Rightarrow \cos xdx=\frac{dt}{2} \\ \therefore I=\frac{1}{2}\int \frac{dt}{t} \\ =\int \frac{1}{2}\log \left|t\right|+C \\ =\frac{1}{2}\log \left|3+2\sin x\right|+C\left(\because t=3+2\sin x\right) \end{gathered}$$