Question

Evaluate the integral
${\int }_0^{{\pi }_{/}2}\frac{secx}{secx+co\sec x}dx$

• A. $\pi /3$
• B. $\pi /2$
• C. $\pi /4$
• D. $\pi /8$

Answer 1

Answer: (C)

$\pi /4$

$I={\int }_0^{\frac{\pi }{2}}\frac{secx}{secx+cosecx}dx$

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

$I={\int }_0^{\frac{\pi }{2}}\frac{secx}{secx+cosecx}dx={\int }_0^{\frac{\pi }{2}}\frac{cosecx}{secx+cosecx}dx=I$

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

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

$I=\frac{\pi }{4}$