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WOLFRAM|DEMONSTRATIONS PROJECT

Euler's Substitutions for the Integral of a Particular Function

Euler's substitutions transform an integral of the form
∫Rx,
a
2
x
+bx+c
dx
, where
R
is a rational function of two arguments, into an integral of a rational function in the variable
t
. Euler's second and third substitutions select a point on the curve
2
y
=a
2
x
+bx+c
according to a method dependent on the parameter values and make
t
the parameter in the parametrized family of lines through that point. Euler's first substitution, used in the case where the curve is a hyperbola, lets
t
be the
y
intercept of a line parallel to one of the asymptotes of the curve. This Demonstration shows these curves and lines.
In symbolic calculations, the Demonstration shows:
1. If
a>0
, the substitution can be
a
2
x
+bx+c
=t±
a
x
. We only consider the case
t-
a
x
.
2. If
a
2
x
+bx+c=a(x-λ)(x-μ)
, where
λ
and
μ
are real numbers, the substitution is
a(x-λ)(x-μ)
=t(x-λ)
.
3. If
c>0
, the substitution can be
a
2
x
+bx+c
=xt±
c
. We only consider the case
xt+
c
.
In all three cases, a linear equation for
x
in terms of
t
is obtained. So
x
,
dx
, and
a
2
x
+bx+c
are rational expressions in
t
.
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