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Integral
calculus |
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The
indefinite integral |
Integrating
rational functions
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The
Ostrogradsky method of the integration of a proper rational
functions
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The
Ostrogradsky method of the integration of a proper rational
functions, examples
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Integrating
rational functions
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Recall
that a rational function is a ratio of two polynomials
written Pn(x)
/ Qm(x).
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To
integrate a rational function
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in
case n >
m, that is, if the
degree of P
is greater than or equal to degree of Q,
we first divide Pn(x)
by Qm(x)
to obtain the sum of the polynomial Pn
-
m(x)
and a proper or simple rational function.
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Then,
we integrate proper rational function using decomposition of rational function
into a sum of partial fractions.
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The
Ostrogradsky method of the integration of a proper rational
functions
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The
integration of a rational fraction whose denominator Q(x)
has multiple or repeated roots.
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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
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where
Q1(x)
is the greatest common divisor of Q(x)
and its derivative Q'
(x),
while
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Q2(x)
= Q (x) / Q1(x).
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Undetermined
coefficients of he polynomials P1(x)
and P2(x),
whose
degrees are one less than of the polynomials
Q1(x)
and Q2(x)
respectively, we calculate by deriving the above integral
identity.
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The
Ostrogradsky method of the integration of a proper rational
functions, examples
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Since
Q(x)
= (x3 + 1)2
and Q'
(x) = 6x2(x3 + 1)
then Q1(x)
= x3 + 1
is the greatest common divisor of Q(x)
and Q'
(x),
and Q2(x)
= Q (x) / Q1(x)
= x3 + 1.
Therefore,
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Since
Q(x)
= (x + 1)2 · (x2 + 1)2
and Q'
(x) = 2(x + 1)
· (x2 + 1)2
+
4x(x + 1)2
· (x2 + 1)
or
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Q'
(x) = 2(x + 1)(x2 + 1)[(x2 + 1)
+ 2x(x + 1)]
then Q1(x)
= (x + 1) (x2 + 1)
and
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Q2(x)
= Q (x) / Q1(x)
= (x + 1) (x2 + 1).
Therefore,
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Calculus contents
F |
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