Question

The value of the integral $\int\limits_{0}^{3\alpha}cosec(x-\alpha)cosec(x-2\alpha)dx$ is

• A. $2\,\sec\,\alpha\,\log\left(\frac{1}{2}cosec\,\alpha\right)$
• B. $2\,\sec\,\alpha\,\log\left(\frac{1}{2}\sec\,\alpha\right)$
• C. $2\,cosec\,\alpha\,\log\left(\sec\,\alpha\right)$
• D. $2\,cosec\,\alpha\,\log\left(\frac{1}{2}\sec\,\alpha\right)$

Answer 1

Answer: (D)

$\int\limits_{0}^{3\alpha} cosec \left(x-a\right) cosec\left(x-2\alpha\right)dx$
$=\int\limits_{0}^{3\alpha} \frac{1}{sin\left(x-\alpha\right)sin\left(x-2\alpha\right)}dx$
$=\int\limits_{0}^{3\alpha} \frac{sin\left[\left(x-\alpha\right)-\left(x-2\alpha\right)\right]}{sin\,\alpha\left[sin\left(x-\alpha\right)sin\left(x-2\alpha\right)\right]} dx$
$=\frac{1}{sin\,\alpha} \int\limits_{0}^{3\alpha}\frac{sin\left(x-\alpha\right)cos\left(x-2\alpha\right)-cos\left(x-\alpha\right)sin\left(x-2\alpha\right)}{sin\left(x-\alpha\right)sin\left(x-2\alpha\right)}dx$
$=\frac{1}{sin\,\alpha}\left[\int\limits_{0}^{3\alpha} \frac{cos\left(x-2\alpha\right)}{sin\left(x-2\alpha\right)} dx-\int\limits_{0}^{3\alpha} \frac{cos\left(x-\alpha\right)}{sin\left(x-\alpha\right)}dx\right]$
$\frac{1}{sin\,\alpha}\left[\left|log\,sin\left(x-2\alpha\right)\right|_{0}^{3\alpha}-\left|log\,sin\left(x-\alpha\right)\right|_{0}^{3\alpha}\right]$
$=\frac{1}{sin\,\alpha}\left[\left(log\,sin\,\alpha-log\,sin\left(-2\alpha\right)\right)\right]-\left[log\left(sin\,2\alpha\right)-log\,sin\left(-\alpha\right)\right]$
$=\frac{1}{sin\,\alpha}\left[log\frac{sin\,\alpha}{sin\,2\alpha}-log\left(\frac{-sin\,2\alpha}{-sin\,\alpha}\right)\right]$
$=\frac{1}{sin\,\alpha}\left[log \frac{sin\,\alpha}{sin\,2\alpha}-log \frac{sin\,2\alpha}{sin\,\alpha}\right]$
$=\frac{1}{sin\,\alpha}\left[log \frac{sin\,\alpha}{2\,sin\,\alpha\,cos\,\alpha}-log\frac{2\,sin\,\alpha\,cos\,\alpha}{sin\,\alpha}\right]$
$=\frac{1}{sin\,\alpha}\left[log \frac{1}{2\,cos\,\alpha}-log\left(2\,cos\,\alpha\right)\right]$
$=\frac{1}{sin\,\alpha}\left[-2\,log\left(2\,cos\,\alpha\right)\right]$
$=2 cosec \alpha\cdot log\left(2\,cos\,\alpha\right)^{-1}$
$=2 cosec \, \alpha\cdot log \frac{1}{2}\left(sec\,\alpha\right)$