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\noindent {\bf Problem statement} One of the physicists' favorite
methods of computing definite integrals is illustrated by the examples
in this problem. It is a {\it very}\/ slick trick. The general idea is:
put in a parameter, differentiate with respect to that parameter, and
see what happens.

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\noindent a) Suppose $\displaystyle { F(a) = \int_0^\infty e^{-ax}\,
dx}$, where $a$ is a positive number. Compute ${F} (a)$ directly. Then
differentiate both sides $N$ times with respect to $a$ (the definite
integral itself and its value!), and set $a=1$. The results are
integral formulas used in statistics (the Gamma ($\Gamma$) function)
and in computing Laplace transforms (with many engineering
applications).

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\noindent b) Suppose $\displaystyle { G(a) = \int_0^\infty
{1\over{x^2+a}}\, dx}$, where $a$ is a positive number.  Compute
$G (a)$ directly. Then differentiate both sides (the definite
integral itself and its value!) $3$ times with respect to $a$. The
result, after a little bit of algebra, will be a formula for
$\displaystyle {\int_0^\infty {1\over {(x^2+a)^4}}\, dx}$. This
formula can be obtained using partial fractions, but this method is
much faster.

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\noindent {\bf Comment} A mathematician might say, ``The justification
of this method is not obvious.''  A physicist might reply, ``But it
works.''










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