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Euler substitutions

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Substitutions of the variable $x=x(t)$ in an integral

\[\tag{1} \int R(x,\sqrt{ax^2+bx+c})\mathrm{d}x,\]

where $R$ 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 $a>0$, 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 (1) 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

\[\tag{2} y^2= ax^2+bx+c \]

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 (2), 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 (2) 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 (2) 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://www.encyclopediaofmath.org/index.php?title=Euler_substitutions&oldid=32131
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article