Question

Write a value of

\[\int\frac{\cos x}{3 + 2 \sin x}\text{  dx}\]

Answer 1

\[\text{ Let I} = \int \frac{\cos x}{3 + 2 \sin x}dx\]
\[\text{ Let 3} + 2 \sin x = t\]
\[ \Rightarrow 2 \text{ cos x dx }= dt\]
\[ \Rightarrow \text{ cos x dx }= \frac{dt}{2}\]
\[ \therefore I = \frac{1}{2}\int\frac{dt}{t}\]
\[ = \int\frac{1}{2} \text{ log }\left| t \right| + C\]
\[ = \frac{1}{2}\text{ log} \left| 3 + 2 \sin x \right| + C \left( \because t = 3 + 2 \text{ sin x} \right)\]