The error function erf(x) = (2)/(√(π)) ∫^x0 e^-t^2 dt is used in probability, statistics, and engineering. (a) Show that ∫^ba e^-t^2 dt = (1)/(2) √(π) [ erf(b) - erf(a) ] . (b) Show that the function y = e^x^2 erf(x) satisfies the differential equation y' = 2xy + 2/√(π) . | Numerade

Consider the error function erf : \mathbb{R} \rightarrow \mathb... | Filo

The error function, erf (x), is defined by erf(x)=(2)/(√(π)) ∫0^x e^-t^2 d t. (a) Let u=erf(x) . Use integration by parts to write ∫erf(x) d x=u v-∫v u^' d x . Give u^' and v^'. (b) Evaluate the integral ∫v u^' d x from part (a) by making a substitution w. Give the values of w and d w. (c) Use your answers to parts (a) and (b) to find ∫erf(x) d x . Your answer may involve erf (x). | Numerade