Question

Find the integral:

Answer 1

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

Answer 2

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Answer 3

what if you took
sqrt(x) = x / sqrt(x)
and u = sqrt(x)

Answer 4

How does it help?

Answer 5

u = sqrt(x) in the exponential
u^2 = x multiplied to the exponential

du = 1/2 sqrt(x) dx , what we put into the denominator, to a constant multiple

Answer 6

Which gives me 2u^3 e^u du

Should I do integration by parts now?

Answer 7

Yea, integration by parts twice should get rid of your u^2 factor (should be u^2, because u = sqrt(x) and x = (sqrt(x))^2 = u^2), the e^u will stay and its integral is very easy to take in the end.

Answer 8

I see. I was wondering if we needed to apply any properties of definite integral and solve it directly.

Answer 9

Hm.. I can't say I know of any specific neat properties that work with the original case. Finding the antiderivative and then substituting in the boundaries was my only idea here.

Answer 10

Okay. Thanks!

Answer 11

I stumbled across something while looking if there was an easier way of going about it (I've run into the x^n e^x case often, but always had to integrate by parts)
If you've never seen it, it was called a 'tabular method of repeated integration by parts' and looked useful for doing these annoying multi-step integration by parts. it is news to me at least, lol.
http://math.ucsd.edu/~wgarner/math20b/int_by_parts.htm

Answer 12

Yes that's new to me too! Thanks for sharing :)