Question

$\underset{0}{\overset{\mathrm{\pi }/3}{\int }}\frac{\cos x}{3+4\sin x}dx$

Answer 1

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