Question

Evaluate the integral

${\int }_0^{\pi /2}\frac{5\tan x-3\cot x}{\tan x+\cot x}dx$

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

Answer 1

Answer: (C)

$\pi /2$

$I={\int }_0^{\frac{\pi }{2}}\frac{5tanx-3cotx}{tanx+cotx}dx$

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

$I={\int }_0^{\frac{\pi }{2}}\frac{5cotx-3tanx}{tanx+cotx}dx$

$2I={\int }_0^{\frac{\pi }{2}}\left(\frac{2tanx+2cotx}{tanx+cotx}\right)dx$

$2I=2{\int }_0^{\frac{\pi }{2}}1\cdot dx$

$I=\frac{\pi }{2}$

Hence, option 'C' is correct.