If ( 5 x+1 ( x-1)( x-2) ^2 x d x ) (=6 |f(x)|+11 |g(x)|+c ), then ((f(x), g(x))= )

Step-by-step solution

(I= 5 x+1 ( x-1)( x-2) ^2 x d x ) Let ( x=t - cosec ^2 x d x=d t ), so ( aligned & I=- 5 t+1 (t-1)(t-2) d t & Let 5 t+1 (t-1)(t-2) = A t-1 + B t-2 & 5 t+1=(A+B) t-(2 A+B) & So, A+B=5 and 2 A+B=-1 & A=-6 and B=11 & I=6 d t t-1 -11 d t t-2 & =6 |t-1|-11 |t-2|+c & =6 | x-1|+11 |( x-2)⁻¹ |+c aligned ) On comparing, we get ((f(x), g(x))= (( x-1),( x-2)⁻¹ ) ) Hence, option (a) is correct.