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Calculus/Integration

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Expert: Scotto - 7/21/2008


Question
How do you evaluate this definite Integral : Integral of (x^2)*e^(-x^2) with limits from 0 to 1?
I tried to use the following technique but I could not do it:
letting u = x^2, du = 2xdx
1/2du = xdx;
substituting everything back is where I got stuck.


Answer
Let u=x, so du=dx and dv=xe^(-x²), so v=e^(-x²)/2.
This problem fits in with the u-v problems where
the integral(udv) = uv - integral(vdu).

Integral(vdu) = integral(e^(-x^2)/2) from 0 to 1.  This integral is almost the normal distribution, which can’t be integrated with bounds on it.  The variables σ and µ are 1 and 0, respectively.  The only difference between that and a normal distribution is that the normal distribution is divided by √(2π).  Find the values on the normal table at 1 (that value a 0 is 0) and multiply by √(2π) to get the result of the integration.

Once you have this value, subtract it from e^(-1/2) (our uv) to get the final answer.

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Scotto

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Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.

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Experience in the area: I have tutored students in all areas of mathematics for over 25 years. Education/Credentials: BSand MS in Mathematics from Oregon State University, where I completed sophomore course in Physics and Chemistry. I received both degrees with high honors. Awards and Honors: I have passed Actuarial tests 100, 110, and 135.

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Maybe not a publication, but I have respond to well oveer 7,500 questions on the PC. Well over 2,000 of them have been in calculus.

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