Question

The value of $\int x\left(cosecxcotx\right)dx=$ is

• A. $xcosecx-log|tanx/2|+c$
• B. $2-xcosecx+\log |tan\frac{x}{2}|+c$
• C. $xcosecx-2\log |tanx/2|+c$
• D. $xcotx-log|tan\frac{x}{2}|+c$

Answer 1

The correct option is B $2-xcosecx+\log |tan\frac{x}{2}|+c$

$\int x\left(cosecxcot\right)dx=x\int \left(cosecxcotx\right)dx$$-\int 1(\int cosecxcotxdx)dx$
$=x[-cosecx]+c+\int \left(cosecx\right)dx$
$=-xcosecx+c+log|cosecx-cotx|$
$=-xcosecx+c+log\left|\frac{1-cosx}{sinx}\right|$
$=2-xcosecx+c_1+log\left|tanx/2\right|$
$\int xcosecxcotxdx$
$=2-xcosecx+c_1+log\left|tan\frac{x}{2}\right|$