MHT CET20225 Aug 2022Morning ShiftMathematicsIndefinite IntegrationActual
x+1 x (1+x e ^x )^2 ~d x= (where C is a constant of integration.)
Options
- A(1+x e ^x )+ 1 1+x e ^x +C
- B( x e ^x 1+x e ^x )+ 1 1+x e ^x +C
- C( x e ^x 1+x e ^x )+C
- D( x e ^x 1+x e ^x )- 1 1+x e ^x +C
Correct answer
B. ( x e ^x 1+x e ^x )+ 1 1+x e ^x +C
Step-by-step solution
aligned & (x+1) d x x (1+x e ^x )^2 = e ^x(x+1) d x x e ^x (1+x e ^x )^2 & = d t t(1+t)^2 [ let x e =t e (x+1) d x= d t] & = 1 t - 1 1+t - 1 (1+t)^2 d t & = |t|- |1+t|+ 1 1+t +C & = ( t 1+t )+ 1 1+t +C & = ( x e ^x 1+x e ^x )+ 1 1+x e ^x +C aligned