Question

How do I integrate this?

Answer 1

\[\Large \int\limits_{0}^{1}e ^{\sqrt{x}}\]

Answer 2

\(\color{black}{\displaystyle \int e^{\sqrt{x}}dx= \int \frac{\sqrt{x}}{\sqrt{x}}e^{\sqrt{x}}dx }\)

Answer 3

\(\color{black}{\displaystyle u=\sqrt{x}}\)
\(\color{black}{\displaystyle du=\frac{1}{2\sqrt{x}}dx}\)

Answer 4

Or, you can rewrite du as
\(\color{black}{\displaystyle 2du=\frac{1}{\sqrt{x}}dx }\)

Answer 5

So, can you write your new integral with this sub?

Answer 6

Whoa, now I get it. Then it would be \[\large 2\int\limits_{0}^{1} ue^u\]

Answer 7

Yes, this is right!

Answer 8

du ... you got it.

Answer 9

So how did you know you should multiply the integrand by \[\frac{ \sqrt{x} }{ \sqrt{x} }\]

Answer 10

I know that the derivative of \(\sqrt{x}=\frac{1}{2\sqrt{x}}\), so you would want to have that derivative inside the integrand, right?

Answer 11

however, we can't just multiply times 1/Sqrt(x), since that changes the integral, so not to violate, we use the "magic 1" technique.

Answer 12

Ah, I see

Answer 13

Thanks for the help!

Answer 14

Yes, and the rest is just integration by parts ...
yw!

Answer 15

(If questions, ask)