Question

${\int }_0^{\frac{\pi }{2}}\frac{3secx+5cosecx}{secx+cosecx}dx=$

• A. $\pi$
• B. $2\pi$
• C. $3\pi$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is B $2\pi$

${\int }_0^{\frac{\pi }{2}}\frac{3secx+5cosecx}{secx+cosecx}dx={\int }_0^{\frac{\pi }{2}}\frac{3sec(\frac{\pi }{2}-x)+5cosec(\frac{\pi }{2}-x)}{secx+cosecx}dx$
$={\int }_0^{\frac{\pi }{2}}\frac{3cosecx+5secx}{secx+cosecx}dx$
$\therefore 2I={\int }_0^{\frac{\pi }{2}}\frac{8secx+8cosecx}{secx+cosecx}dx=8{\int }_0^{\frac{\pi }{2}}1dx=8[x{]}_0^{\frac{\pi }{2}}=8\frac{\pi }{2}=4\pi$