Question

The value of $\displaystyle\int_{0}^{\pi/2}\log|\tan\,x+\cot\,x|dx$ is

• A. $\pi\,\log\,2$
• B. $-\pi\,\log\,2$
• C. $\frac{\pi}{2}\log\,2$
• D. $-\frac{\pi}{2}\log\,2$

Answer 1

Answer: (A)

$\int\limits_{0}^{\pi/ 2} log \left|tan\,x+cot\,x\right|dx$
$=\int\limits_{0}^{\pi/ 2} log \left|\frac{sin^{2}\,x+cos^{2}\,x}{cos\,x\,sin\,x}\right|dx$
$=\int\limits_{0}^{\pi /2} log \left(\frac{1}{sin\,x cos\,x}\right)dx$
[$\because$ $sin\,x, cos\,x are +Ve$ in the first quadrant]
$=\int\limits_{0}^{\pi /2} log1 \, dx-\int\limits_{0}^{\pi /2} log sin\,x\, dx-\int\limits_{0}^{\pi /2} log \, cos\,x\,dx$
$=0-\left(-\frac{\pi}{2}log\,2\right)-\left(-\frac{\pi}{2}log\,2\right)$
$=\frac{\pi}{2} log\,2+\frac{\pi}{2} log\,2=\pi\, log\,2$