Euler substitutions

Substitutions of the variable in an integral

(1)

where 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 , then

The second Euler substitution: If the roots and of the quadratic polynomial are real, then

The third Euler substitution: If , then

(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 and to be expressed rationally in terms of the new variable .

The first two Euler substitutions permit the reduction of (1) to the integral of a rational function over any interval on which takes only real values.

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

(2)

has a rational parametric representation; for if is chosen to be the angular coefficient of the pencil of straight lines passing through a point of (2), then the coordinates of any point on this curve can be expressed rationally in terms of . In the case when , that is, when (2) is a hyperbola, the first Euler substitution is obtained by taking as one of the points at infinity defined by the directions of the asymptotes of this hyperbola; when the roots and of the quadratic polynomial are real, the second Euler substitution is obtained by taking as one of the points or ; finally, when , the third Euler substitution is obtained by taking as one of the points where the curve (2) intersects the ordinate axis, that is, one of the points .