Question

The integral ${\int }_{7\pi /4}^{7\pi /3}\sqrt{tn{a}^{2}xdx}$ is equal to:

• A. $log2\sqrt{2}$
• B. $log2$
• C. $2log2$
• D. $log\sqrt{2}$

Answer 1

Answer: (D)

Let $I=\underset{7\pi /4}{\overset{7\pi /3}{\int }}$$\sqrt{ta{n}^{2}xdx}$
$I=\underset{7\pi /4}{\overset{7\pi /3}{\int }}$$tanxdx=-logcosx{|}_{7\pi /4}^{7\pi /3}$
$=\left[logcos\frac{7\pi }{3}-logcos\frac{7\pi }{4}\right]$
$=logcos\frac{7\pi }{4}-logcos\frac{7\pi }{3}$
$=log\left[\frac{cos\frac{7\pi }{4}}{cos\frac{7\pi }{4}}\right]=log\left[\frac{cos\left(2\pi -\frac{\pi }{4}\right)}{cos\left(2\pi +\frac{\pi }{3}\right)}\right]$
$=log\left[\frac{cos\frac{\pi }{4}}{cos\frac{\pi }{3}}\right]=log\left(\frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}}\right)$
$log\left(\frac{2}{\sqrt{2}}\right)=log\sqrt{2}.$