Euler's Substitutions for the Integral of a Particular Function
Euler's Substitutions for the Integral of a Particular Function
Euler's substitutions transform an integral of the form , where is a rational function of two arguments, into an integral of a rational function in the variable . Euler's second and third substitutions select a point on the curve =a+bx+c according to a method dependent on the parameter values and make 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 be the intercept of a line parallel to one of the asymptotes of the curve. This Demonstration shows these curves and lines.
∫Rx,
a+bx+c
dx2
x
R
t
2
y
2
x
t
t
y
In symbolic calculations, the Demonstration shows:
1. If , the substitution can be . We only consider the case .
a>0
a+bx+c
=t±2
x
a
xt-
a
x2. If , where and are real numbers, the substitution is .
a+bx+c=a(x-λ)(x-μ)
2
x
λ
μ
a(x-λ)(x-μ)
=t(x-λ)3. If , the substitution can be . We only consider the case .
c>0
a+bx+c
=xt±2
x
c
xt+
c
In all three cases, a linear equation for in terms of is obtained. So , , and are rational expressions in .
x
t
x
dx
a+bx+c
2
x
t