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<h1>The Gaussian integral</h1>
<p>The integral <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\int_{-\infty}^{\infty} e^{-x^2}\,dx</annotation></semantics></math></span> appears throughout
probability.<sup class="footnote-ref"><a href="#fn1" id="fnref1">1</a></sup></p>
<div class="math display"><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mtext> </mtext><mi>d</mi><mi>x</mi><mo>=</mo><msqrt><mi>π</mi></msqrt></mrow><annotation encoding="application/x-tex">\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}</annotation></semantics></math></span></div>
<h2>Two proofs</h2>
<ol>
<li><strong>Polar coordinates</strong>: square the integral.</li>
<li><em>Feynman's trick</em>: differentiate under the integral.</li>
</ol>
<table>
<thead>
<tr>
<th>Method</th>
<th style="text-align:right">Steps</th>
</tr>
</thead>
<tbody>
<tr>
<td>Polar</td>
<td style="text-align:right">3</td>
</tr>
<tr>
<td>Series</td>
<td style="text-align:right">5</td>
</tr>
</tbody>
</table>
<section class="footnotes">
<ol>
<li id="fn1">See also <a href="https://en.wikipedia.org/wiki/Gaussian_integral">https://en.wikipedia.org/wiki/Gaussian_integral</a>. <a href="#fnref1" class="footnote-back">↩</a></li>
</ol>
</section>