Step-by-step explanation:
∫ erf(x) dx
Integrate by parts.
If u = erf(x), then du = 2/√π e^(-x²) dx.
If dv = dx, then v = x.
∫ u dv = uv − ∫ v du
= x erf(x) − ∫ 2/√π x e^(-x²) dx
= x erf(x) + 1/√π ∫ -2x e^(-x²) dx
Substitution.
If u = -x², then du = -2x dx.
= x erf(x) + 1/√π ∫ e^u du
= x erf(x) + 1/√π e^u + C
= x erf(x) + 1/√π e^(-x²) + C
