MHT CET202519 Apr 2025Morning ShiftMathematicsIndefinite IntegrationActual
d x 2 e ^ 2 x +3 e ^x+1 =
Options
- Ax+ ( e ^x+1 )-2 (2 e ^x+1 )+ c , where c is the constant of integration
- Bx- ( e ^x+1 )+4 ( e ^x+1 )+ c where c is the constant of integration
- Cx+ ( e ^x+1 )-4 (2 e ^x+1 )+ c , where c is the constant of integration
- Dx- ( e ^x+1 )+2 (2 e ^x+1 )+ c , where c is the constant of integration
Correct answer
A. x+ ( e ^x+1 )-2 (2 e ^x+1 )+ c , where c is the constant of integration
Step-by-step solution
Evaluate the integral I = d x 2 e ^ 2x +3 e ^x+1 . Substitute with t = e ^x , so d t = e ^x d x and d x = d t t . The integral transforms to I = d t t(2t^2+3t+1) . Factor the denominator: 2t^2+3t+1 = (2t+1)(t+1) , yielding I = d t t(t+1)(2t+1) . Perform partial fraction decomposition: 1 t(t+1)(2t+1) = A t + B t+1 + C 2t+1 . Solving gives A=1 , B=1 , and C=-4 , so 1 t + 1 t+1 - 4 2t+1 . Integrate: I = 1 t d t + 1 t+1 d t - 4 1 2t+1 d t = |t| + |t+1| - 2 |2t+1| + c . Substitute back: t = e ^x > 0 , so I = x + ( e ^x+