Integrate by parts e^(sqrt(x))?

Question

Integrate by parts: Integral of e^(sqrt(x)) dx or e^(x^(1/2)) dx. Thanks!

Answer 1

First, I will do a u-substitution, but I will use 'a' instead of 'u' (because I will be doing integration by parts later, which also uses a 'u')

let a = Sqrt[x]

da = 1 / (2 * Sqrt[x]) dx => 2 * Sqrt[x] da = dx => 2 a da = dx

So the Integral now becomes:

Integral of e^Sqrt[x] dx

Integral of e^a * 2 a da

2 * Integral of e^a a da

Integrating by parts now, with the formula:

Integral of u dv = u v - Integral of v du

using:

u = a

dv = e^a da

du = da

v = e^a

Thus I can just substitute in the values for 'u' and 'v'

2 (u v - Integral of v du)

2 (a e^a - Integral of e^a da)

2 (a e^a - e^a + C)

Now all that is left is replacing 'a' with it's value in terms of x and simplifying.

2 (Sqrt[x] e^Sqrt[x] - e^Sqrt[x] + C)

2 e^Sqrt[x] (Sqrt[x] - 1) + C

That should be the final answer (also, Mathematica confirms that it is correct :D )