MHT CET202312 May 2023Morning ShiftMathematicsIndefinite IntegrationActual
The integral ^2 x ^2 x ( ^5 x+ ^3 x ^2 x+ ^3 x ^2 x+ ^5 x )^2 ~d x is equal to
Options
- A1 3 (1+ ^3 x ) + c , where c is a constant of integration.
- B-1 3 (1+ ^3 x ) + c , where c is a constant of integration.
- C1 1+ ^3 x + c , where c is a constant of integration.
- D-1 1+ ^3 x + c , where c is a constant of integration.
Correct answer
B. -1 3 (1+ ^3 x ) + c , where c is a constant of integration.
Step-by-step solution
Let aligned I & = ^2 x ^2 x ( ^5 x+ ^3 x ^2 x+ ^3 x ^2 x+ ^5 x )^2 ~d x & = ^2 x ^2 x ( ^5 x+ ^3 x ^2 x+ ^5 x ^2 x+ ^5 x )^2 ~d x & = ^2 x ^2 x [ ^3 x ( ^2 x+ ^2 x )+ ^3 x ( ^2 x+ ^2 x ) ]^2 ~d x & = ^2 x ^2 x ( ^3 x+ ^3 x )^2 ~d x & = ^2 x ^2 x (1+ ^3 x )^2 ~d x aligned [Dividing numerator and denominator by ^6 x ] Let 1+ ^3 x=t Differentiating w.r.t. x , we get aligned & 3 ^2 x ^2 x ~d x= dt & ^2 x ^2 x ~d x= 1 3 dt aligned ^2 x ^2 x ~d x= 1 3 dt aligned I & = 1 3 1 t ^2 dt & = -1 3 t + c & = -1 3 (1+ ^3 x ) + c