MHT CET202520 Apr 2025Evening ShiftMathematicsIndefinite IntegrationActual
e^x (x-1) (x+1)^3 ~d x=
Options
- Ae ^x(x+1)^2+ c , where c is the constant of integration
- Be ^x(x+1)^3+ c , where c is the constant of integration
- Ce ^x (x+1)^2 + c , where c is the constant of integration
- De ^x (x+1)^3 + c , where c is the constant of integration
Correct answer
C. e ^x (x+1)^2 + c , where c is the constant of integration
Step-by-step solution
Evaluate the integral e^x x-1 (x+1)^3 d x using the standard formula e^x [f(x) + f'(x)] d x = e^x f(x) + C . Express the integrand in the appropriate form by rewriting the numerator: x-1 (x+1)^3 = (x+1) - 2 (x+1)^3 = 1 (x+1)^2 - 2 (x+1)^3 Let f(x) = 1 (x+1)^2 Then f'(x) = - 2 (x+1)^3 Thus the integrand becomes f(x) + f'(x) Applying the formula yields: e^x [ 1 (x+1)^2 - 2 (x+1)^3 ] d x = e^x 1 (x+1)^2 + C = e^x (x+1)^2 + C This corresponds to option C .