(x+1) x(1+xe^x)^2 dx =

Step-by-step solution

Let I = x+1 x(1+xe^x)^2 dx Multiplying the numerator and the denominator by e^x , we get: I = (x+1)e^x xe^x(1+xe^x)^2 dx Let xe^x = t . Differentiating both sides with respect to x , we get: (1 e^x + x e^x) dx = dt (x+1)e^x dx = dt Substituting these into the integral: I = dt t(1+t)^2 Using partial fractions: 1 t(1+t)^2 = 1+t-t t(1+t)^2 = 1 t(1+t) - 1 (1+t)^2 1 t(1+t)^2 = 1+t-t t(1+t) - 1 (1+t)^2 = 1 t - 1 1+t - 1 (1+t)^2 Integrating with respect to t : I = ( 1 t - 1 1+t - 1 (1+t)^2 ) dt I = |t| - |1+t| + 1 1+t + c