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$$\int^{\infty}_{0} e^{-x}x^{100}dx$$

I am sure is something here I can not see, else it is integration by parts 100 times.

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    $\begingroup$ Use this answer: math.stackexchange.com/questions/20397/… $\endgroup$ – user940 Feb 14 '16 at 15:08
  • $\begingroup$ @ByronSchmuland is there a proof? $\endgroup$ – gbox Feb 14 '16 at 15:11
  • $\begingroup$ Yes. The proof is "repeated integration by parts". $\endgroup$ – user940 Feb 14 '16 at 15:17
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Hint Consider the integral $$I(n) := \int_0^{\infty} x^n e^{-x} dx$$ (NB with a general exponent $n$ of $x$). Applying integration by parts with $u = x^n$, $dv = e^{-x} dx$, gives $$\int_0^{\infty} x^n e^{-x} dx = \left. x^n e^{-x} \right\vert_0^{\infty} + n \int_0^{\infty} x^{n - 1} e^x .$$ We can show that the first term is zero (for example, with an inductive argument and L'Hopital's Rule), and so $$\int_0^{\infty} x^n e^{-x} dx = n \int_0^{\infty} x^{n - 1} e^x ,$$ which we can write suggestively as the recurrence relation $$I(n) = n I(n - 1).$$

So, we can evaluate our original integral, which by definition is $I(100)$, by writing $$I(100) = 100 \cdot I(99) = 100 \cdot 99 \cdot I(98) = \cdots.$$ On the other hand, it's easy to compute that $I(0) = 1$.

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Use a trick which, if I remember correctly, is due to Feynman:

For $\alpha > 0 $, let $$I(\alpha) = \int_{0}^{\infty} e^{-\alpha x} dx = \frac{1}{\alpha}$$

Differentiating under the integral sign $n$ times (with respect to $\alpha$) gives $$\int_{0}^{\infty} (-1)^nx^ne^{-\alpha x}dx = (-1)^n\frac{n!}{\alpha^n} $$ whence $$\int_{0}^{\infty} x^ne^{-\alpha x}dx = \frac{n!}{\alpha^n}. $$ Taking $n=100$ and $\alpha = 1$, shows that the sought integral is equal to $100!$.

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Notice that $\Gamma(t) = \int\limits_0^\infty \mathrm{e}^{-x}x^{t-1} \,\mathrm{d}x$ is the gamma function. Therefore, $$ \int\limits_0^\infty \mathrm{e}^{-x}x^{100} \,\mathrm{d}x = \Gamma(101) = 100! $$

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This is the Gamma function of $101$. Indeed by definition:

$$\Gamma[x] = \int_0^{+\infty} t^{x-1}\ e^{-t}\ \text{d}t$$

And so

$$\int_0^{+\infty} t^{100}\ e^{-t}\ \text{d}t = \Gamma[101] = 100!$$

About the integration

This integration can be done in several ways. By parts is surely the most intuitive one, even if it may be tedious (although you need just two-three passages to guess the whole behavior of the integration).

Otherwise, we can do it with the Feynman trick as shown below/above.

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