COMEDK2023Morning ShiftMathematicsIndefinite IntegrationActual
x d x 2(1+x)^ 3 / 2 is equal to
Options
- Ax 1+x +C
- B2+x x 1+x +C
- C- x 1+x +C
- D2+x 1+x +C
Correct answer
D. 2+x 1+x +C
Step-by-step solution
Let I = x 2(1+x)^ 3/2 dx . Substitute u = 1+x , so du = dx and x = u-1 . Substituting these into the integral: I = 1 2 u-1 u^ 3/2 du = 1 2 (u^ -1/2 - u^ -3/2 ) du . Integrating with respect to u : I = 1 2 [ u^ 1/2 1/2 - u^ -1/2 -1/2 ] + C = 1 2 [2 u + 2 1 u ] + C = u + 1 u + C . Substituting u = 1+x back: I = 1+x + 1 1+x + C = 1+x+1 1+x + C = 2+x 1+x + C . Answer: 2+x 1+x +C