
Integral
calculus 

The
indefinite integral 
Integrating
rational functions

The
Ostrogradsky method of the integration of a proper rational
functions

The
Ostrogradsky method of the integration of a proper rational
functions, examples






Integrating
rational functions

Recall
that a rational function is a ratio of two polynomials
written P_{n}(x)
/ Q_{m}(x).

To
integrate a rational function


in
case n >
m, that is, if the
degree of P
is greater than or equal to degree of Q,
we first divide P_{n}(x)
by Q_{m}(x)
to obtain the sum of the polynomial P_{n
}_{}_{
m}(x)
and a proper or simple rational function.

Then,
we integrate proper rational function using decomposition of rational function
into a sum of partial fractions.


The
Ostrogradsky method of the integration of a proper rational
functions

The
integration of a rational fraction whose denominator Q(x)
has multiple or repeated roots.

If
P(x)
and Q(x)
are polynomials with real coefficients and P(x)
/ Q(x)
is a proper fraction, and Q(x)
has multiple
roots, then


where
Q_{1}(x)
is the greatest common divisor of Q(x)
and its derivative Q'
(x),
while

Q_{2}(x)
= Q_{ }(x) / Q_{1}(x).

Undetermined
coefficients of he polynomials P_{1}(x)
and P_{2}(x),
whose
degrees are one less than of the polynomials
Q_{1}(x)
and Q_{2}(x)
respectively, we calculate by deriving the above integral
identity.


The
Ostrogradsky method of the integration of a proper rational
functions, examples


Since
Q(x)
= (x^{3} + 1)^{2}
and Q'
(x) = 6x^{2}(x^{3} + 1)^{}
then Q_{1}(x)
= x^{3} + 1
is the greatest common divisor of Q(x)
and Q'
(x),
and Q_{2}(x)
= Q_{ }(x) / Q_{1}(x)
= x^{3} + 1.
Therefore,





Since
Q(x)
= (x + 1)^{2 }· (x^{2} + 1)^{2}
and Q'
(x) = 2(x + 1)^{
}· (x^{2} + 1)^{2}
+
4x(x + 1)^{2
}· (x^{2} + 1)
or

Q'
(x) = 2(x + 1)(x^{2} + 1)[(x^{2} + 1)
+ 2x(x + 1)]
then Q_{1}(x)
= (x + 1) (x^{2} + 1)
and

Q_{2}(x)
= Q_{ }(x) / Q_{1}(x)
= (x + 1) (x^{2} + 1).
Therefore,











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