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Calculus/integrate e

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Expert: - 9/5/2008


Question
Hello,

How do I integrate e^(square root of x) with respect to x?  Thank you.

Answer
To do this requires a u substitution, addition of some expressions, and then a u-dv substitution.  I'm sure after looking at it, a shorter way could be found, but this is straight forward.

Do a u-substitution with u=√x, du=dx/(2√x); multiply the integral by (2√x)/(2√x).  The denominator and the dx convert to du, the numerator converts to 2u, and e^√x converts to e^u.

The final result is integrate(ue^udu).  Convert the u to a y so that you have integral(ye^y dy).  You are now ready for a u-v integeral.
You know, integral(u dv) = uv - integral(v du).

Let u=y, dv=e^y, so du=dy and v=e^y dy.  The varibles on the right are uv, which you can get by plugging in the expressions for u and v.  The integral of (v du) can also be found by putting in what v and du are.  The integral(e^y dy) is just e^y.

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