I can't complete the integration of $e^{\sqrt{x}}$

Question

Compute $\displaystyle\int_0^1e^{\sqrt{x}}\,dx$

That's a picture of how far I could get while trying to integrate $e^{\sqrt{x}}$. I tried the substitution method first, (boxed part) and then went for the parts one, the problem is that when I want to find $V$, I need to integrate the starting function, so in the end I'm looped. Maybe I was doing something wrong, thanks for the help before hand.

Answer 1

Hint. One may just make the change of variable $$ u=\sqrt{x},\quad x=u^2,\quad dx=2udu $$ giving $$ \int e^{\sqrt{x}}dx=2\int ue^udu $$ which may be evaluated by parts, thus

$$ \int e^{\sqrt{x}}dx=2(\sqrt{x}-1)e^{\sqrt{x}}+C, $$

for any constant $C$.

Answer 2

Consider the change of variable $\sqrt x = t$. This turns tour integral into

$$\int \limits _0 ^1 \Bbb e ^{\sqrt x} \ \Bbb d x = \int \limits _0 ^1 \Bbb e ^t 2t \ \Bbb d t$$

which can now be integrated by parts:

$$2 \int \limits _0 ^1 \Bbb e ^t t \ \Bbb d t = 2 t \Bbb e^t \Big| _0 ^1 - 2 \int \limits _0 ^1 \Bbb e ^t = 2 \Bbb e - 2 \Bbb e ^t \Big| _0 ^1 = 2 \Bbb e - (2 \Bbb e - 2) = 2 .$$