Question

$\int cosec (x- a) cosec \: x \: dx $ is equal to

• A. $\frac {1}{\sin\,a}\log|\sin(x-a)\sin\,x|+C$
• B. $\frac {-1}{\sin\,a}\log|\sin\, x\, cosec(x-a)|+C$
• C. $\frac {-1}{\sin\,a}\log|\sin(x-a)\sin\,x|+C$
• D. $-\frac{1}{\sin a} \log |\sin (x-a) \operatorname{cosec} x|+C$

Answer 1

Answer: (B)

Let $l=\int \operatorname{cosec}(x-a) \operatorname{cosec} x d x$
$=\int \frac{\sin a}{\sin a \sin (x-a) \sin x} d x$
$=-\frac{1}{\sin a} \int \frac{\sin [(x-a)-x]}{\sin (x-a) \sin } d x$
$=-\frac{1}{\sin a} \int\left[\frac{\sin (x-a) \cos x-\cos (x-a) \sin x}{\sin (x-a) \sin x}\right] d x$
$=-\frac{1}{\sin a} \int[\cot x-\cot (x-a] d x$
$=-\frac{1}{\sin a}[\log |\sin x|-\log |\sin (x-a)|]+C$
$=-\frac{1}{\sin a}[\log || \sin x \operatorname{cosec}(x-a) \mid]+C$