Question

If $\int \frac{sinx}{sin(x-a)}dx=Ax+Blogsin(x-a)+C$ then values of $(A,B)$ is

• A. $(sin\alpha ,cos\alpha )$
• B. $(cos\alpha ,sin\alpha )$
• C. $(-sin\alpha ,cos\alpha )$
• D. $(-cos\alpha ,sin\alpha )$

Answer 1

The correct option is B $(cos\alpha ,sin\alpha )$

Consider the given integral.

$I=\int \frac{\sin x}{\sin (x-a)}dx$

$I=\int \frac{\sin (x-a+a)}{\sin (x-a)}dx$

$I=\int \frac{\sin (x-a)\cos a+\cos (x-a)\sin a}{\sin (x-a)}dx$

$I=\int (\cos a+\cot (x-a\left)\sin a\right)dx$

$I=x\cos a+\sin alog\sin (x-a)+C$

Therefore,

$A=\cos a,B=\sin a$

Hence, this is the answer.