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\noindent {\bf Problem statement} a) The improper integral converges:
${\int_0^{\infty} x e^{-x^2} \, dx}$. What is its value?

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\noindent b) The value of ${\int_0^{\infty} e^{-x^2} \, dx}$, another
convergent improper integral, is ${{\sqrt \pi} \over 2}$. This amazing
fact is easiest to explain with some of the tools in third semester
calculus. Improper integrals involving polynomials and $e^{-x^2}$
often arise in statistics and therefore in analysis of
experiments. Use integration by parts to get a formula relating
${\int_0^{\infty} x^n e^{-x^2} \, dx}$ and
${\int_0^{\infty} x^{n-2} e^{-x^2} \, dx}$, where $n$ is
a positive integer bigger than 2. (The parts to take are slightly
tricky.)

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\noindent c) Now find the values of

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\settabs 3 \columns
\+i) ${\int_0^{\infty} x^2 e^{-x^2} \, dx}$
&ii) ${\int_0^{\infty} x^3 e^{-x^2} \, dx}$
&iii) ${\int_0^{\infty} x^4 e^{-x^2} \, dx}$\cr

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\noindent You will need the reduction formula in b) and the two
initial values found in a) and b).










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