Integrating
irrational functions using Euler's
substitutions
Integrating
irrational functions using Euler's substitutions examples
Example:Evaluate
As the leading coefficient of the
quadratic a
> 0 we can use first Euler's
substitution, therefore
Solution:
Example:Evaluate
As the leading coefficient a
and vertical translation y0
of the quadratic have different signs, i.e., a · y0<
0
the polynomial can be factorized using its real roots, hence we
use second Euler's substitution,
Solution:
Example:Evaluate
Since the constant (or free) term
of the given quadratic
c > 0, we
use third Euler's substitution, thus
Solution:
Binomial
integral
Integral
of the form
is
called the binomial integral where, a
and b
are real numbers while m,
n
and p
are rational numbers.
If
m,
n
and p
all are integers then the integrand is a rational function
integration of which is shown above.
There
are only three cases the binomial integral can be solved by
elementary functions:
1. if
m
and n
are
fractions and p
is an integer then, the integral can be solved using
substitution x
= t s, where s
is the least common denominator of m
and n.
2.
if p
is a fraction
and (m +
1) / n is an
integer, then the integral can be solved using substitution
a + bxn = t s,
where s
is denominator of p.
3.
if p
is a fraction
and (m +
1) / n + p
is an integer then, the integral can be solved using substitution
ax - n
+ b = t s,
where s
is denominator of p.