Question

The integral ${\int }_0^{{\pi }_{/}3}\frac{\cos x}{3+4\sin x}{d}_{X=}$

• A. $\log \left(\frac{3+2\sqrt{3}}{3}\right)$
• B. $\frac{1}{4}\log \left(\frac{3+2\sqrt{3}}{3}\right)$
• C. $2\log \left(\frac{3+2\sqrt{3}}{3}\right)$
• D. $\frac{1}{2}\log \left(\frac{3+2\sqrt{3}}{2}\right)$

Answer 1

Answer: (B)

$\frac{1}{4}\log \left(\frac{3+2\sqrt{3}}{3}\right)$

${\int }_0^{\frac{\pi }{3}}\frac{\cos x}{3+4\sin x}dx={\int }_0^{\frac{\pi }{3}}\frac{d\left(\sin x\right)}{3+4\sin x}$

$={\int }_0^{\frac{\pi }{3}}\frac{d\left(\sin x\right)}{3+4\sin x}=\frac{1}{4}{\left[\log \right(3+4\sin x\left)\right]}_0^{\frac{\pi }{3}}$

$=\frac{1}{4}[\log [3+4\left(\sin \frac{\pi }{3}\right)]-\log (3+4\sin \left(0\right)\left)\right]$

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

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

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

$\therefore {\int }_0^{\frac{\pi }{3}}\frac{\cos x}{3+4\sin x}dx=\frac{1}{4}\log \left(\frac{3+2\sqrt{3}}{3}\right)$