COMEDK2024Evening ShiftMathematicsIndefinite IntegrationActual
e^x [ x^2+1 (x+1)^2 ] d x is equal to
Options
- Ae^x ( x-1 x+1 )+C
- Be^x x+1 +C
- Cx e^x x+1 +C
- D- e^x x+1 +C
Correct answer
A. e^x ( x-1 x+1 )+C
Step-by-step solution
The integral is of the form e^x [f(x) + f'(x)] dx = e^x f(x) + C . We rewrite the integrand as follows: x^2 + 1 (x + 1)^2 = x^2 - 1 + 2 (x + 1)^2 = (x - 1)(x + 1) + 2 (x + 1)^2 = x - 1 x + 1 + 2 (x + 1)^2 Let f(x) = x - 1 x + 1 . Then f'(x) = (x + 1)(1) - (x - 1)(1) (x + 1)^2 = x + 1 - x + 1 (x + 1)^2 = 2 (x + 1)^2 . Since the integrand is e^x [f(x) + f'(x)] , the integral is e^x f(x) + C . Therefore, the integral is e^x ( x - 1 x + 1 ) + C . Answer: e^x ( x-1 x+1 )+C