MHT CET20249 May 2024Evening ShiftMathematicsIndefinite IntegrationActual
The value of x+1 x (1+x e^x )^2 ~d x is equal to
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 )- x 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= e ^x(x+1) e ^x x (1+x e ^x )^2 ~d x & & Put x e ^x= t & (x+1) e ^x ~d x= dt & & I = d t (1+ t )^2 aligned aligned & = 1+ t - t t (1+ t )^2 dt & = 1 dt t (1+ t ) - 1 (1+ t )^2 dt & = 1+ t - t t (1+ t ) - 1 (1+ t )^2 dt & = 1 t dt - 1 ( t +1) dt - 1 (1+ t )^2 dt & = 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