Integration by parts of Erf(x)

Question

I need to integrate from 0-t. I am told to set u= Erf and u'= (2e^-x^2)/pi^1/2.  v' is 1 and v=x.

i am also told that the answer is t Erf(t)+ ((e^-t^2)-1)/pi^1/2

 

how many times do I need to integrate by parts to come to that answer? It gets more complicated to me each time I integrate, no matter which assignments I use. Can I somehow do it keeping the Erf designation or do I need to use the actual function (which they didn't give in this assignment.)

 

TIA

Answer 1

The integration-by-parts formula is

∫u v' dx = uv - ∫u' v dx

 

so with u =erf(x), u' = 2e^-x²/√π, v' = 1, and v = x, you get

∫erf(x) dx =  x erf(x) - (2/√π) ∫xe^-x² dx = x erf(x) + (1/√π) e^-x² ,

where I used a u-substitution (u=e^-x²) to evaluate the right-hand-side integral.

 

Now just evaluate this from 0 to t:

∫₀^t erf(x) dx = t erf(t) + (1/√π) (e^-t² - 1) .