MHT CET202525 Apr 2025Morning ShiftMathematicsIndefinite IntegrationActual
e ^x ( x+5 (x+6)^2 ) d x is
Options
- Ae ^x (x+6)^2 + c , where c is the constant of integration.
- Be^x x+5 +c, where c is the constant of integration.
- Ce ^x (x+5)^2 + c , where c is the constant of integration.
- De ^x x+6 + c , where c is the constant of integration.
Correct answer
D. e ^x x+6 + c , where c is the constant of integration.
Step-by-step solution
Evaluation of the integral e^x ( x+5 (x+6)^2 ) , dx relies on recognizing it as e^x (f(x) + f'(x)) , dx , which integrates to e^x f(x) + c . Express the integrand as x+5 (x+6)^2 = (x+6)-1 (x+6)^2 = 1 x+6 - 1 (x+6)^2 . Let f(x) = 1 x+6 , so f'(x) = - 1 (x+6)^2 . The integrand matches e^x (f(x) + f'(x)) , yielding the integral. Thus, e^x ( 1 x+6 - 1 (x+6)^2 ) , dx = e^x 1 x+6 + c . e^x ( 1 x+6 ) + c corresponds to option D. Final answer: D