MHT CET202314 May 2023Morning ShiftMathematicsIndefinite IntegrationActual
x+ ^3 x 2 x ~d x= A x+ B f (x)+ c (where c is a constant of integration). Then values of A, B and f (x) are
Options
- AA = 1 2 , ~B = -3 4 2 , f (x)= 2 x-1 2 x+1
- BA=- 1 2 , B= -3 4 2 , f(x)= 2 x+1 2 x-1
- CA = 1 2 , ~B = -3 4 2 , f (x)= 2 x+1 2 x-1
- DA = 3 2 , ~B = 1 2 , f (x)= 2 x-1 2 x+1
Correct answer
A. A = 1 2 , ~B = -3 4 2 , f (x)= 2 x-1 2 x+1
Step-by-step solution
Let aligned I & = x+ ^3 x 2 x ~d x & = x (1+ ^2 x ) 2 x ~d x & = x (1+1- ^2 x ) 2 ^2 x-1 ~d x & = x (2- ^2 x ) 2 ^2 x-1 ~d x aligned Put x= t x ~d x=- dt I =- 2- t ^2 2 t ^2-1 dt aligned & = t^2-2 2 t^2-1 d t & = 1 2 2 t^2-4 2 t^2-1 d t & = 1 2 (1- 3 2 t^2-1 ) d t & = 1 2 d t- 3 2 d t ( 2 t)^2-1^2 & = 1 2 t- 3 2 1 2 1 2 | 2 t-1 2 t+1 |+c & = 1 2 x- 3 4 2 | 2 x-1 2 x+1 |+ c A A & = 1 2 , ~B = -3 4 2 , f (x)= 2 x-1 2 x+1 aligned