Question

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

Answer 1

\[Let\ I = \int_0^\frac{\pi}{3} \frac{\cos x}{3 + 4 \sin x} d x . \]
\[Let\ \sin\ x\ = t . Then, \cos\ x\ dx\ = dt\]
\[When\ x = 0, t = 0\ and\ x\ = \frac{\pi}{3}, t = \frac{\sqrt{3}}{2}\]
\[ \therefore I = \int_0^\frac{\pi}{3} \frac{\cos x}{3 + 4\sin x} d x\]
\[ = \int_0^\frac{\sqrt{3}}{2} \frac{1}{3 + 4t} d t\]
\[ = \frac{1}{4} \left[ \log \left( 3 + 4t \right) \right]_0^\frac{\sqrt{3}}{2} \]
\[ = \frac{1}{4}\left( \log \left( 3 + 2\sqrt{3} \right) - \log 3 \right)\]
\[ = \frac{1}{4} \log \left( \frac{3 + 2\sqrt{3}}{3} \right)\]