Question

Integrate the rational function: \(\frac {cos \ x}{(1-sin\ x)(2-sin\ x)}\)

Hint:

Put sin x = t

Answer 1

\(\frac {cos \ x}{(1-sin\ x)(2-sin\ x)}\)

\(Let \ sin \ x = t ⇒ cos \ x\  dx = dt\)

∴ \(∫\)\(\frac {cos \ x}{(1-sin\ x)(2-sin\ x)}\ dx\)= \(∫\frac {dt}{(1-t)(2-t)}\)

\(Let\)  \(\frac {1}{(1-t)(2-t)}\) = \(\frac {A}{(1-t)}+\frac {B}{(2-t)}\)

\(1 = A(2-t)+B(1-t)  \)                       \(  ...(1)\)

\(Substituting\  t = 2 \ and \ then\  t = 1 \ in \ equation \ (1), we\  obtain\)

\(A = 1\  and\  B = −1\)

∴ \(\frac {1}{(1-t)(2-t)}\) = \(\frac {1}{(1-t)}-\frac {1}{(2-t)}\)

⇒ \(∫\)\(\frac {cos \ x}{(1-sin\ x)(2-sin\ x)}\ dx\) = \(∫\)\([\frac {1}{(1-t)}-\frac {1}{(2-t)}]\ dt\)

= \(-log\ |1-t|+log\ |2-t|+C\)

= \(log\ |\frac {2-t}{1-t}|+C\)

= \(log\ |\frac {2-sin\ x}{1-sin\ x}|+C\)