Question

${\int }_0^{\pi /2}(2\tan \frac{x}{2}+x{\sec }^2\frac{x}{2})dx=$

• A. $\pi$
• B. $\pi /2$
• C. 2$\pi /3$
• D. $\pi /6$

Answer 1

The correct option is A $\pi$

${\int }_0^{\frac{\pi }{2}}(2tan\frac{x}{2}+xse{c}^2\frac{x}{2})dx$
$=2{\int }_0^{\frac{\pi }{2}}(tan\frac{x}{2}+\frac{1}{2}xse{c}^2\frac{x}{2})dx$
$=2{\int }_0^{\frac{\pi }{2}}d\left(xtan\frac{x}{2}\right)$
$=2\left(xtan\frac{x}{2}\right){\int }_0^{\frac{\pi }{2}}$
$=2[\left(\frac{\pi }{2}1\right)-0]$
$=\pi$