Question

${\int }_{\pi /3}^{\pi /3}\frac{x\sin x}{{\cos }^{2}x}dx$ is equal to

• A. $\frac{\pi }{3}-\log \tan \frac{3\pi }{2}$
• B. $2[\frac{2\pi }{3}-\log \tan \frac{5\pi }{12}]$
• C. $3[\frac{\pi }{2}\log \sin \frac{\pi }{12}]$
• D. none of these

Answer 1

Answer: (B)

$\underset{-\pi /3}{\overset{\pi /3}{\int }}\frac{xsinx}{co{s}^{2}x}dx=2$
$\underset{0}{\overset{\pi /3}{\int }}x\left(secxtanx\right)dx$
$=2\left[{\left|x\cdot secx\right|}_0^{\pi /3}-{\int }_0^{\pi /3}secxdx\right]$
$=2\left[\frac{\pi }{3}\cdot \left(2\right)-{\left|log\left(secx+tanx\right)\right|}_0^{\pi /3}\right]$
$=2\left[\frac{2\pi }{3}-log\left(2+\sqrt{3}\right)+0\right]$
$=2\left[\frac{2\pi }{3}-logtan\frac{5\pi }{12}\right]$
$\left[\because tan\frac{5\pi }{12}=tan75^{\circ }=2+\sqrt{3}\right]$