Expert: Paul Klarreich - 3/22/2010

Question
how would you integrate e^x^1/2dx using a u substitution of u = x^1/2 then integration by parts?

Answer
Questioner: Keith
Country: United States
Category: Calculus
Private: No
Subject: intergration by parts with u substitution
Question: how would you integrate e^x^1/2dx using a u substitution of u = x^1/2 then integration by parts?


{
| e^(x^1/2) dx
}

I don't like to use 'u' if I am going to do IBP, so let's use 'z' instead:

Use  z = x^1/2,  z^2 = x,  2z dz = dx, and you have:

{
| 2 e^z z dz
}

Now try IBP:  u = z,  dv = e^z dz   <<  z is easy to diff,  e^z is easy to integrate.

du = dz,  v = e^z

Now you have your:

uv part is  z e^z

MINUS:

{
| e^z dz
}  

You should be able to finish up now. (don't forget to put back the '2', and to substitute back to get x's.