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Math 152-copyright Joe Kahlig, 14C
Section 8.4: Integration of Rational Functions by Partial Fractions
3
4
3(x + 5) + 4(x + 2)
7x + 23
+
=
= 2
x+2 x+5
(x + 2)(x + 5)
x + 7x + 10
Z
7x + 23
dx =
x2 + 7x + 10
Z
3
4
+
dx =
x+2 x+5
P (x)
where both P (x) and Q(x) are polynomials. The
A rational function is a function of the form Q(x)
degree of a polynomial is the highest power of the variable.
P (x)
NOTE: To integrate a rational function, Q(x)
, with the partial fraction method, you MUST HAVE
the degree P (x) < degree Q(x). If this is not the case then use long division(or some other method)
to find J(x) and K(x) so that
P (x)
K(x)
= J(x) +
Q(x)
Q(x)
Method of Integration by Partial Fractions:
1) Factor the denominator completely
2) Decompose the fraction
3) Solve for the constants in the decomposition
4) Integrate the new fractions
Z
x3 + 2x2 − 5
dx
x+1
Z
x3 + 3x − 5
dx
x2 + 1
1.
2.
Math 152-copyright Joe Kahlig, 14C
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Write out the partial fraction decomposition. Do not determine the numerical values of the coefficients.
3.
x3
5x + 4
+ 3x2 − 10x
4.
x−3
x(x + 1)3 (x2 + 5)
5.
x2 + 2
(x2 − 9)(x2 + 16)2
Math 152-copyright Joe Kahlig, 14C
Compute these integrals.
Z
6.
x3
Z
7.
5x + 4
dx
+ 3x2 − 10x
x+2
dx
x3 + 2x
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Math 152-copyright Joe Kahlig, 14C
Z
8.
15x + 5
dx
(x + 2)2 (x2 + 1)
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