dx Take cos‌x=t then −cosec2xdx=dt This implies, ∫
5t+1
(t−1)(t−2)
dt=
A
t−1
+
B
t−2
Then, 5t+1=(A+B)t−(2A+B) So,A+B=5 and 2A+B=−1 This gives, A=−6,B=11 Then, I=6‌∫
dt
t−1
−11‌∫
dt
t−2
=6‌log|t−1|−11‌log|t−2|+c =6‌log|cot‌x−1|+11‌log|(cot‌x−2)−1|+c On comparison, (f(x),g(x))=((cot‌x−1),(cot‌‌x−2)−1)