MHT CET202617 April 2026Morning ShiftMathematicsIndefinite IntegrationActual
The value of 2x^ 1 3 [3] x^2 ,dx is
Options
- A(x^ 2 3 x^ 2 3 + x^ 2 3 ) + c
- B3 [x^ 2 3 x^ 2 3 - x^ 2 3 ] + c
- C3 [-x^ 2 3 x^ 2 3 - x^ 2 3 ] + c
- D3 [-x^ 2 3 x^ 2 3 + x^ 2 3 ] + c
Correct answer
D. 3 [-x^ 2 3 x^ 2 3 + x^ 2 3 ] + c
Step-by-step solution
Let x^ 2 3 = t . Differentiating with respect to x , we get 2 3 x^ - 1 3 dx = dt dx = 3 2 x^ 1 3 dt . Substituting this in the integral: I = 2x^ 1 3 t ( 3 2 x^ 1 3 dt ) I = 3x^ 2 3 t , dt Since x^ 2 3 = t , we have: I = 3 t t , dt Using integration by parts: I = 3 [ t(- t) - (- t) , dt ] I = 3 [ -t t + t ] + c Substituting t = x^ 2 3 back: I = 3 [ -x^ 2 3 x^ 2 3 + x^ 2 3 ] + c Answer: 3 [-x^ 2 3 x^ 2 3 + x^ 2 3 ] + c