Question

Evaluate: ∫ dx/ sin (x – a) cos(x – b)

Answer 1

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

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

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

\(= \frac 1{\cos (a - b)} \int \frac{\cos (x - b)\cos(x- a) + \sin(x - b)\sin(x - a)}{\sin(x - a)\cos(x -b)} dx\)

\(= \frac 1{\cos (a - b)} \int \cot (x -a)dx+ \frac 1{\cos(a - b)} \int \tan(x - b)dx\)

\(= \frac 1{\cos (a - b)} [\log|\sin(x -a)| - \log|\cos(x - b)|] +C\)

\(= \frac 1{\cos (a - b)} \log\left|\frac{\sin (x -a)}{\cos(x - b)}\right| + C\)

Answer 2

The given integral is

= (1/ cos(b – a)) ∫ (cot(x – a) + tan(x – b)) dx 

= (1/ cos(b – a)) [log |sin(x – a)| – log (sec{x – b})] + c 

= (1/ cos(b – a)) [log |sin(x – a)cos(x – b)|] + c