MHT CET202411 May 2024Evening ShiftMathematicsIndefinite IntegrationActual
x+1 x (1+x e ^x )^2 ~d x=
Options
- A| x e ^x 1+x e ^x |+ x 1+x e ^x + c , (where c is a constant of integration)
- B| x e ^x 1+x e ^x |+ e ^x 1+x e ^x + c (where c is a constant of integration)
- C| x e ^x 1+x e ^x |+ 1 1+x e ^x + c , (where c is a constant of integration)
- D| x e ^x 1+x e ^x |- 1 1+x e ^x + c , (where c is a constant of integration)
Correct answer
C. | x e ^x 1+x e ^x |+ 1 1+x e ^x + c , (where c is a constant of integration)
Step-by-step solution
aligned & Let I = x+1 x (1+x e ^x )^2 ~d x & = (x+1) e ^x x e ^x (1+x e ^x )^2 ~d x & Let x e ^x= t & (e^x+x e ^x ) d x= dt & I= d t t(1+t)^2 & = (1+ t )- t t (1+ t )^2 dt & = 1 t d t- 1 (1-t) d t- 1 (1+t)^2 d t aligned aligned & = | t |- |1+ t |+ 1 (1+ t ) + c & = | x e ^x 1+x e ^x |+ 1 (1+x e ^x ) + c aligned