Question

Evaluate: $\int \frac{dx}{\cos (x-a)\cos (x-b)}$

• A. $\frac{1}{\sin (a-b)}\log |\frac{\sec (x-a)}{\sec (x-b)}|+c$
• B. $\frac{1}{\sin (a+b)}\log |\frac{\cos (x-b)}{\cos (x-a)}|+c$
• C. $\frac{1}{\sin (b-a)}\log |\frac{\cos (x-b)}{\cos (x-a)}|+c$
• D. $\frac{1}{\sin (a+b)}\log |\frac{\cos (x-a)}{\cos (x-b)}|+c$

Answer 1

The correct option is C $\frac{1}{\sin (b-a)}\log |\frac{\cos (x-b)}{\cos (x-a)}|+c$

$\frac{1}{\sin (b-a)}\int \frac{\sin \left[\right(x-a)-(x-b\left)\right]}{\cos (x-a)\cos (x-b)}dx$

$\Rightarrow \frac{1}{\sin (b-a)}\int \frac{\sin (x-a)\cos (x-b)-\sin (x-b)\cos (x-a)}{\cos (x-a)\cos (x-b)}dx$

$\Rightarrow \frac{1}{\sin (b-a)}\int \left(\tan \right(x-a)dx-\int \tan (x-b\left)\right)dx$

$\Rightarrow \frac{1}{\sin (b-a)}\left(\log |\sec \right(x-a)|-\log |\sec (x-b\left)|\right)+c$

$\Rightarrow \frac{1}{\sin (b-a)}\log \left|\frac{\sec (x-a)}{\sec (x-b)}\right|+c$

$\Rightarrow \frac{1}{\sin (b-a)}\log \left|\frac{\cos (x-b)}{\cos (x-a)}\right|+c$