# 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