I need help with an Integral (Substitution).

Question

anonymous · Answer

$$\int\limits_{0}^{100}e^\sqrt{x}dx=?$$
I am supposed to solve this using the method of substitution. 
I have no clue how the hell this would help in that weird case.

hartnn · Answer

u know integration by parts ?

across · Answer

Let$$
u=\sqrt x.\tag{1}$$Then,$$
du=\frac1{2\sqrt x}dx\Longrightarrow dx=2\sqrt xdu=2udu.\tag{2}
$$It follows from $(1)$ that$$
x=0\Longrightarrow u=0\text{, and}\
x=100\Longrightarrow u=10.
$$So, from $(1)$, $(2)$ and the new limits, we have that$$
2\int_0^{10}ue^u\,du.
$$

anonymous · Answer

@hartnn Yes, that would be my first impulse but I should use the substitution method.

hartnn · Answer

as shown by @across

anonymous · Answer

Ok, and after that I have to integrate by parts do integrate the $$ue^udu$$
part, right?

hartnn · Answer

yes.

across · Answer

Yes, or you could use the (not so) common knowledge that$$
\int xe^x\,dx=e^x(x-1).
$$

anonymous · Answer

^^ I thought that I saw that Integral before, but not today it seems. Thx for your help.

hartnn · Answer

more general formula :
$\large \int e^x(f(x)+f'(x))dx=e^xf(x)+c$