Euler substitutions

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 $(0,\pm\sqrt{c})$.