Expert: Ahmed Salami - 1/30/2005

Question
The question is first make a substitution and then use integration by parts to evaluate the integral.

integral(from 4-1)e^sqrt(x)dx

I'm not sure if my technique is correct.  

Let w=sqrt(x)
then, x=w^2
dx=2wdw

I now have
integral(4-1)(e^w)2wdw

let u=2w
du=2dw
dv=e^w
v=e^w

integral(4-1)2w(e^w)dw=2w(e^w)-integral(e^w)2dw
=[2w(e^w)-2(e^w)](4-1)

=[2sqrt(x)(e^(sqrt(x)))-2(e^sqrt(x))](4-1)
=[2sqrt(4)(e^sqrt(4))-2e^sqry(4)]-[2sqrt(1)e^sqrt(1)-2e^sqrt(1)]
=(4e^2-2e^2)-(2e-2e)
=2e^2

If you could just let me know if my math is making sense or if I'm going wrong anyware.  
Thank You

Answer
Hi Buffy,
Sorry for the delay.
I'll have to say you've done really well and got the right result.
But it would have been easier to integrate by the substitution x = u^2
dx = 2u du
integral e^sqrtx dx becomes integral 2ue^u du
Integrating by parts gives
2[ue^u - integral e^u du]
= 2(ue^u - e^u)
= 2e^u(u-1)
= 2e^sqrtx (sqrtx - 1)
Introducing the limits i.e from 1 to 4 gives 2e^2
You can always get back to me.
Regards.