MHT CET20239 May 2023Morning ShiftMathematicsIndefinite IntegrationActual
x+1 x (1+x e ^x )^2 d x=
Options
- A| x e ^x 1+x e ^x |+ c , where c is a constant of integration.
- B| x e ^x 1+x e ^x |- 1 1+x e ^x + c , where c is a constant of integration.
- C|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
D. | 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= e ^x(x+1) e ^x x (1+x e ^x )^2 ~d x & Let x e ^x= t & (x+1) e ^x ~d x= dt & I = dt t (1+ t )^2 & = 1+ t - t t (1+ t )^2 dt & = 1 d t t(1+t) - 1 (1+t)^2 d t & = 1+t-t t(1+t) - 1 (1+t)^2 d t & = 1 t d t- 1 (t+1) d t- 1 (1+t)^2 d t & = t - dt ( t +1) - 1 (1+ t )^2 dt + c & Let y=1+ t & d y= dt & I = t - 1 y ~d y- 1 y^2 ~d y+ c & = t - y+ 1 y + c & = t - (1+ t )+ 1 (1+ t ) + c & = x e ^x- (1+x e ^x )+ 1 (1+x e ^x ) + c & I = | x e ^x 1+x e ^x |+ 1 1+x e ^x + c & aligned