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Substitutions of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e0365901.png" /> in an integral
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Substitutions of the variable $x=x(t)$ in an integral
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\begin{equation}
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\label{eq1} \int R(x,\sqrt{ax^2+bx+c})\mathrm{d}x,
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\end{equation}
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where $R$ is a rational function of its arguments, that reduce \eqref{eq1} to the integral of a rational function. There are three types of such substitutions.
  
<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>
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The first Euler substitution: If $a>0$, then
  
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.
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\[ \sqrt{ax^2+bx+c}=\pm x\sqrt{a}\pm t.\]
  
The first Euler substitution: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e0365904.png" />, then
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The second Euler substitution: If the roots $x_1$ and $x_2$ of the quadratic polynomial $ax^2+bx+c$ 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/e0365905.png" /></td> </tr></table>
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\[\sqrt{ax^2+bx+c}=\pm t(x-x_1).\]
  
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
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The third Euler substitution: If $c>0$, 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>
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\[ \sqrt{ ax^2+bx+c} = \pm \sqrt{c}\pm xt.\]
  
The third Euler substitution: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036590/e03659010.png" />, then
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(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 $x$ and $\sqrt{ax^2+bx+c}$ to be expressed rationally in terms of the new variable $t$.
  
<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>
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The first two Euler substitutions permit the reduction of \eqref{eq1} to the integral of a rational function over any interval on which $\sqrt{ax^2+bx+c}$ takes only real values.
 
 
(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
 
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>
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\begin{equation}\label{eq2} y^2= ax^2+bx+c \end{equation}
  
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" />.
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has a rational parametric representation; for if $t$ is chosen to be the angular coefficient of the pencil of straight lines $y-y_0=t(x-x_0)$ passing through a point $(x_0,y_0)$ of \eqref{eq2}, then the coordinates of any point on this curve can be expressed rationally in terms of $t$. In the case when $a>0$, that is, when \eqref{eq2} is a hyperbola, the first Euler substitution is obtained by taking as $(x_0,y_0)$ one of the points at infinity defined by the directions of the asymptotes of this hyperbola; when the roots $x_1$ and $x_2$ of the quadratic polynomial $ax^2+bx+c$ are real, the second Euler substitution is obtained by taking as $(x_0,y_0)$ one of the points $(x_1,0)$ or $(x_2,0)$; finally, when $c>0$, the third Euler substitution is obtained by taking as $(x_0,y_0)$ one of the points where the curve \eqref{eq2} intersects the ordinate axis, that is, one of the points $(0,\pm\sqrt{c})$.

Latest revision as of 06:13, 10 April 2023

Substitutions of the variable $x=x(t)$ in an integral \begin{equation} \label{eq1} \int R(x,\sqrt{ax^2+bx+c})\mathrm{d}x, \end{equation} where $R$ is a rational function of its arguments, that reduce \eqref{eq1} to the integral of a rational function. There are three types of such substitutions.

The first Euler substitution: If $a>0$, then

\[ \sqrt{ax^2+bx+c}=\pm x\sqrt{a}\pm t.\]

The second Euler substitution: If the roots $x_1$ and $x_2$ of the quadratic polynomial $ax^2+bx+c$ are real, then

\[\sqrt{ax^2+bx+c}=\pm t(x-x_1).\]

The third Euler substitution: If $c>0$, then

\[ \sqrt{ ax^2+bx+c} = \pm \sqrt{c}\pm xt.\]

(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 $x$ and $\sqrt{ax^2+bx+c}$ to be expressed rationally in terms of the new variable $t$.

The first two Euler substitutions permit the reduction of \eqref{eq1} to the integral of a rational function over any interval on which $\sqrt{ax^2+bx+c}$ takes only real values.

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

\begin{equation}\label{eq2} y^2= ax^2+bx+c \end{equation}

has a rational parametric representation; for if $t$ is chosen to be the angular coefficient of the pencil of straight lines $y-y_0=t(x-x_0)$ passing through a point $(x_0,y_0)$ of \eqref{eq2}, then the coordinates of any point on this curve can be expressed rationally in terms of $t$. In the case when $a>0$, that is, when \eqref{eq2} is a hyperbola, the first Euler substitution is obtained by taking as $(x_0,y_0)$ one of the points at infinity defined by the directions of the asymptotes of this hyperbola; when the roots $x_1$ and $x_2$ of the quadratic polynomial $ax^2+bx+c$ are real, the second Euler substitution is obtained by taking as $(x_0,y_0)$ one of the points $(x_1,0)$ or $(x_2,0)$; finally, when $c>0$, the third Euler substitution is obtained by taking as $(x_0,y_0)$ one of the points where the curve \eqref{eq2} intersects the ordinate axis, that is, one of the points $(0,\pm\sqrt{c})$.

How to Cite This Entry:
Euler substitutions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_substitutions&oldid=14688
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article