Question

Evaluate the integral

${\int }_{\pi /4}^{\pi /2}log(1+cotx)dx$

• A. $\frac{\pi }{4}\log 2$
• B. $\frac{\pi }{8}$ $\log 2$
• C. $\pi \log 2$
• D. $0$

Answer 1

The correct option is B $\frac{\pi }{8}$ $\log 2$

$I={\int }_{\pi {/}_2}^{\pi {/}_4}log(1+cotx)dx$

$I={\int }_{\pi {/}_2}^{\pi {/}_4}log[1+cot(\frac{3\pi }{4}-x)]dx$

$2I={\int }_{\pi {/}_2}^{\pi {/}_4}log\left[\right(1+cotx\left)\right(1+cot(\frac{\pi }{4}-x)]dx$

$2I={\int }_{\pi {/}_2}^{\pi {/}_4}log\left(2\right)$

$2I=\frac{\pi }{4}log\left(2\right)$

$I=\frac{\pi }{8}log2$