Integrate the function \frac{\cos \left ( x+a \right )}{\sin \left ( x+b \right )} W.r.t.x.

 

 

 

 
 
 
 
 

Answers (1)

I= \int \frac{\cos \left ( x+a \right )}{\sin \left ( x+b \right )}\, dx\; \; = \int \frac{\cos \left \{ \left ( x+b \right )+\left ( a-b \right ) \right \}}{\sin \left ( x+b \right )}\: dx
   I= \int \left ( \frac{\cos \left ( x+b \right )\cos \left ( a-b \right )-\sin \left ( x+b \right )\sin \left ( a-b \right )}{\sin \left ( x+b \right )} \right )\, dx
   \Rightarrow \int \left ( \frac{\cos \left ( x+b \right )\cos \left ( a-b \right )}{\sin \left ( x+b \right )}-\frac{\sin \left ( x+b \right )\sin \left ( a-b \right )}{\sin \left ( x+b \right )} \right )\, dx
\Rightarrow \int\left \{ \cos \left ( a-b \right ) \cot \left ( x+b \right )-\sin \left ( a-b \right )\right \}\, dx
\Rightarrow \cos \left ( a-b \right )\log \left | \sin \left ( x+b \right ) \right |-x\: \sin \left ( a-b \right )+c

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