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# Integration by parts of Erf(x)

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

### 1 Answer by Expert Tutors

Andre W. | Friendly tutor for ALL math and physics coursesFriendly tutor for ALL math and physics ...
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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:
0t erf(x) dx = t erf(t) + (1/√π) (e-t² - 1) .