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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
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1 Answer

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) .

Comments

Andre,
Thanks for answering. I got to the point of making the next u assignment e^-x^2. That would make v'=x and v = x^2/2 correct? And it gets a bit messy for me from that point. Can you show those steps? 
Judith,
You only do integration by parts once. The new integral you get, ∫xe-x² dx, can be solved with the u-substitution (no v's!):
u = e-x², du = -2x e-x² ⇒ ∫xe-x² dx = -(1/2) ∫du = -(1/2) u = -(1/2) e-x².
Note that the (-1/2) then cancels with the (-2) out front.
Which we did, when we used the formula at the beginning : ∫erf(x) dx = x erf(x) - (2/√π) ∫xe-x² dx. This is the integration-by-parts part! :)
Right, and I will go with it if I have to but I don't believe that's what is required. They taught the value of integrating with derivatives to avoid complicated, repeated integration by parts. I think they are trying to drive that point home.
That's exactly what we did: we integrated the derivative of e-x², thereby avoiding a second integration by parts.

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