Question

Evaluate the integral
${\int }_0^{\pi /2}\frac{3\sec x+5\text{cosec}x}{\sec x+\text{cosec}x}dx$

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

Answer 1

Answer: (B)

$2\pi$

$I={\int }_0^{\frac{\pi }{2}}\left(\frac{3\sec x+5\text{cosec}x}{\sec x+\text{cosec}x}\right)dx$

${\int }_0^{a}f\left(x\right)dx={\int }_o^{a}f(a-x)dx$

$I={\int }_0^{\frac{\pi }{2}}\left(\frac{3\text{cosec}x+5\sec x}{\sec x+\text{cosec}x}\right)dx$

$2I={\int }_0^{\pi /2}8\left(1\right)dx$

$I=(8\cdot \frac{\pi }{2})\cdot \frac{1}{2}$

$I=2\pi$