MHT CET202615 April 2026Evening ShiftMathematicsIndefinite IntegrationActual
The value of ^6 x + ^6 x ^2 x ^2 x , d x is
Options
- Ax + ^2 x + c , where c is the constant of integration
- Bx - x - 3x + c , where c is the constant of integration
- C2 ^5 x + 2 ^5 x + c , where c is the constant of integration
- D^3 x + 2 ^3 x + c , where c is the constant of integration
Correct answer
B. x - x - 3x + c , where c is the constant of integration
Step-by-step solution
Let I = ^6 x + ^6 x ^2 x ^2 x , dx Using the algebraic identity a^3 + b^3 = (a+b)(a^2 - ab + b^2) , we can simplify the numerator: ^6 x + ^6 x = ( ^2 x + ^2 x)( ^4 x - ^2 x ^2 x + ^4 x) Since ^2 x + ^2 x = 1 , this becomes: ^4 x - ^2 x ^2 x + ^4 x = ( ^2 x + ^2 x)^2 - 3 ^2 x ^2 x = 1 - 3 ^2 x ^2 x Substituting this back into the integral: I = 1 - 3 ^2 x ^2 x ^2 x ^2 x , dx I = ( 1 ^2 x ^2 x - 3 ) , dx Using the identity 1 = ^2 x + ^2 x in the numerator of the first term: I = ( ^2 x + ^2 x ^2 x ^2 x - 3 ) , dx I = (