Question

\[\int\limits_0^{\pi/3} \frac{\cos x}{3 + 4 \sin x} dx\]

Answer 1

\[\int_0^\frac{\pi}{3} \frac{\cos x}{3 + 4\sin x} d x\]

\[Let, \sin x = t \Rightarrow \cos x dx = dt\]

\[\text{When, }\sin x \to 0 ; t \to 0\]

\[\text{And }\sin x \to \frac{\pi}{3} ; t \to \frac{\sqrt{3}}{2}\]

\[ = \int_0^\frac{\sqrt{3}}{2} \frac{dt}{3 + 4t}\]

\[ = \frac{1}{4}\log \left[ 3 + 4t \right]_0^\frac{\sqrt{3}}{2} \]

\[ = \frac{1}{4}\log\left[ \log\left( 3 + 2\sqrt{3} \right) - \log\left( 3 + 0 \right) \right]\]

\[ = \frac{1}{4}\log\left[ \log\left( 2\sqrt{3} + 3 \right) - \log\left( 3 \right) \right]\]

\[ = \frac{1}{4}\left( \log\frac{2\sqrt{3} + 3}{3} \right)\]