View source for Euler substitutions

Substitutions of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e0365901.png" /> in an integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e0365902.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e0365903.png" /> is a rational function of its arguments, that reduce (1) to the integral of a rational function. There are three types of such substitutions.

The first Euler substitution: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e0365904.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e0365905.png" /></td> </tr></table>

The second Euler substitution: If the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e0365906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e0365907.png" /> of the quadratic polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e0365908.png" /> are real, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e0365909.png" /></td> </tr></table>

The third Euler substitution: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659010.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659011.png" /></td> </tr></table>

(Any combination of signs may be chosen on the right-hand side in each case.) All the Euler substitutions allow both the original variable of integration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659013.png" /> to be expressed rationally in terms of the new variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659014.png" />.

The first two Euler substitutions permit the reduction of (1) to the integral of a rational function over any interval on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659015.png" /> takes only real values.

Geometrically, the Euler substitutions mean that the second-order curve

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

has a rational parametric representation; for if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659017.png" /> is chosen to be the angular coefficient of the pencil of straight lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659018.png" /> passing through a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659019.png" /> of (2), then the coordinates of any point on this curve can be expressed rationally in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659020.png" />. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659021.png" />, that is, when (2) is a hyperbola, the first Euler substitution is obtained by taking as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659022.png" /> one of the points at infinity defined by the directions of the asymptotes of this hyperbola; when the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659024.png" /> of the quadratic polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659025.png" /> are real, the second Euler substitution is obtained by taking as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659026.png" /> one of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659027.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659028.png" />; finally, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659029.png" />, the third Euler substitution is obtained by taking as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659030.png" /> one of the points where the curve (2) intersects the ordinate axis, that is, one of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659031.png" />.