x + 1 x(1 + x e^x) dx =

Step-by-step solution

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