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SECTION 7.4 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS |||| 473

35.

	1	2
Prove the formula A 2 r			for the area of a sector of
Prove the formula A 2 r			for the area of a sector of

a circle with radius r and central angle . [Hint: Assume
0	2 and place the center of the circle at the origin

so it has the equation x 2 y 2 r 2. Then A is the sum of the area of the triangle POQ and the area of the region PQR in the ﬁgure.]

39. (a) Use trigonometric substitution to verify that

y sa 2 t 2 dt 12 a 2 sin 1 x a 12 x sa 2 x 2

(b)Use the ﬁgure to give trigonometric interpretations of both terms on the right side of the equation in part (a).

y	P
	P
O	¨
O	Q	R	x

;36. Evaluate the integral

y x 4 sxdx2 2

Graph the integrand and its indeﬁnite integral on the same screen and check that your answer is reasonable.

;37. Use a graph to approximate the roots of the equation

x 2 s4 x 2 2 x. Then approximate the area bounded by the curve y x 2 s4 x 2 and the line y 2 x.

38.A charged rod of length L produces an electric ﬁeld at point P a, b given by

L a	b
E P y a		dx
	4 0 x 2 b 2 3 2

where is the charge density per unit length on the rod and0 is the free space permittivity (see the ﬁgure). Evaluate the integral to determine an expression for the electric ﬁeld E P .

y	P(a,b)
y


0		x
0	L	x

ay=œ„„„„„a@-t@

0 x t

40.The parabola y 12 x 2 divides the disk x 2 y 2 8 into two parts. Find the areas of both parts.

41.Find the area of the crescent-shaped region (called a lune) bounded by arcs of circles with radii r and R. (See the ﬁgure.)

42.A water storage tank has the shape of a cylinder with diameter 10 ft. It is mounted so that the circular cross-sections are vertical. If the depth of the water is 7 ft, what percentage of the total capacity is being used?

43.A torus is generated by rotating the circle

x 2 y R 2 r 2 about the x-axis. Find the volume

enclosed by the torus.

7.4 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

In this section we show how to integrate any rational function (a ratio of polynomials) by expressing it as a sum of simpler fractions, called partial fractions, that we already know how to integrate. To illustrate the method, observe that by taking the fractions 2 x 1 and 1 x 2 to a common denominator we obtain

2	1		2 x 2 x 1	x 5

x 1		x 2	x 1 x 2	x2 x 2

If we now reverse the procedure, we see how to integrate the function on the right side of

474 \|\|\|\| CHAPTER 7 TECHNIQUES OF INTEGRATION
this equation:
	x 5	2		1
y		dx y				dx
y	x2 x 2	dx y	x 1		x 2	dx

2 ln x 1 ln x 2 C

To see how the method of partial fractions works in general, let’s consider a rational function

f x P x Q x

where P and Q are polynomials. It’s possible to express f as a sum of simpler fractions provided that the degree of P is less than the degree of Q. Such a rational function is called proper. Recall that if

P x an xn an 1xn 1 a1 x a0

≈+x +2 x-1)˛ +x

˛-≈

≈+x ≈-x 2x

2x-2

where an 0, then the degree of P is n and we write deg P n.

If f is improper, that is, deg P deg Q , then we must take the preliminary step of dividing Q into P (by long division) until a remainder R x is obtained such that deg R deg Q . The division statement is

1	f x	P x	S x	R x
		Q x		Q x

where S and R are also polynomials.

As the following example illustrates, sometimes this preliminary step is all that is required.

x3 x

V EXAMPLE 1 Find y x 1 dx.

SOLUTION Since the degree of the numerator is greater than the degree of the denominator, we ﬁrst perform the long division. This enables us to write


3		dx y		x2 x 2			2	dx
y	x x
	x 1						x 1
			x3		x2
						2x 2 ln x 1			C	M
			3		2

The next step is to factor the denominator Q x as far as possible. It can be shown that any polynomial Q can be factored as a product of linear factors (of the form ax b) and irreducible quadratic factors (of the form ax2 bx c, where b2 4ac 0). For instance, if Q x x4 16, we could factor it as

Q x x2 4 x2 4 x 2 x 2 x2 4

The third step is to express the proper rational function R x Q x (from Equation 1) as a sum of partial fractions of the form

A	or	Ax B
ax b i		ax2 bx c j

N Another method for ﬁnding A, B, and C is given in the note after this example.

SECTION 7.4 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS |||| 475

A theorem in algebra guarantees that it is always possible to do this. We explain the details for the four cases that occur.

CASE I N The denominator Q(x) is a product of distinct linear factors.

This means that we can write

Q x a1 x b1 a2 x b2 ak x bk

where no factor is repeated (and no factor is a constant multiple of another). In this case the partial fraction theorem states that there exist constants A1, A2, . . . , Ak such that

2	R x		A1		A2		Ak
	Q x		a1 x b1		a2 x b2		ak x bk

These constants can be determined as in the following example.

x2 2x 1

V EXAMPLE 2 Evaluate y 2x3 3x2 2x dx.

SOLUTION Since the degree of the numerator is less than the degree of the denominator, we don’t need to divide. We factor the denominator as

2x3 3x2 2x x 2x2 3x 2 x 2x 1 x 2

Since the denominator has three distinct linear factors, the partial fraction decomposition of the integrand (2) has the form

3	x2 2x 1		A		B		C
	x 2x 1 x 2		x		2x 1		x 2

To determine the values of A, B, and C, we multiply both sides of this equation by the product of the denominators, x 2x 1 x 2 , obtaining

4 x2 2x 1 A 2x 1 x 2 Bx x 2 Cx 2x 1

Expanding the right side of Equation 4 and writing it in the standard form for polynomials, we get

5	x2 2x 1 2A B 2C x2 3A 2B C x 2A

The polynomials in Equation 5 are identical, so their coefﬁcients must be equal. The coefﬁcient of x2 on the right side, 2A B 2C, must equal the coefﬁcient of x2 on the left side—namely, 1. Likewise, the coefﬁcients of x are equal and the constant terms are equal. This gives the following system of equations for A, B, and C:

2A B 2C 1

3A 2B C 2

2A 2B 2C 1

476 |||| CHAPTER 7 TECHNIQUES OF INTEGRATION

Solving, we get A 12 , B 15 , and C 101 , and so

N We could check our work by taking the terms

x2 2x 1

to a common denominator and adding them.

dx y

2x3 3x2 2x

2x 1

x 2

N Figure 1 shows the graphs of the integrand

2 ln x

ln 2x

1 10

ln x 2 K

in Example 2 and its indeﬁnite integral (with

K 0). Which is which?

In integrating the middle term we have made the mental substitution u 2x 1, which

gives du 2 dx and dx du 2.

We can use an alternative method

to ﬁnd the coefﬁcients A, B, and C in

NOTE

Example 2. Equation 4 is an identity; it is true for every value of x. Let’s choose values of

x that simplify the equation. If we put x 0 in Equation 4, then the second and third terms

on the right side vanish and the equation then becomes 2A 1, or A 21 . Likewise,

x 21 gives 5B 4 41 and x 2 gives 10C 1, so B 51 and C

. (You may object

that Equation 3 is not valid for x 0, 21 , or 2, so why should Equation 4 be valid for those

FIGURE 1

values? In fact, Equation 4 is true for all values of x, even x 0, 21 , and 2. See Exercise 69

for the reason.)

EXAMPLE 3 Find y

, where a 0.

SOLUTION The method of partial fractions gives

x2 a2

x a x a

x a

and therefore

A x a B x a 1

Using the method of the preceding note, we put x a in this equation and get

A 2a 1, so A 1 2a . If we put x a, we get B 2a 1, so B 1 2a .

Thus

x2 a2

x a

(ln

x a

ln x a ) C

Since ln x ln y ln x y , we can write the integral as

x a

x2 a2

x a

See Exercises 55–56 for ways of using Formula 6.

CASE 11 N Q(x) is a product of linear factors, some of which are repeated.

Suppose the ﬁrst linear factor a1 x b1 is repeated r times; that is, a1 x b1 r occurs in the factorization of Q x . Then instead of the single term A1 a1 x b1 in Equation 2, we

N Another method for ﬁnding the coefﬁcients: Put x 1 in (8): B 2.

Put x 1: C 1.

Put x 0: A B C 1.

SECTION 7.4

INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS |||| 477

would use

a1 x b1

a1 x b1 2

a1 x b1 r

By way of illustration, we could write

x3 x 1

x2 x 1 3

x 1

x 1 2

x 1 3

but we prefer to work out in detail a simpler example.


EXAMPLE 4 Find y		x4 2x2 4x 1		dx.
EXAMPLE 4 Find y		x3 x2 x 1		dx.
SOLUTION The ﬁrst step is to divide. The result of long division is
	x4 2x2 4x 1		x 1		4x
		x3 x2 x 1	x 1		x3 x2 x 1
		x3 x2 x 1			x3 x2 x 1

The second step is to factor the denominator Q x x3 x2 x 1. Since Q 1 0, we know that x 1 is a factor and we obtain

x3 x2 x 1 x 1 x2 1 x 1 x 1 x 1x 1 2 x 1

Since the linear factor x 1 occurs twice, the partial fraction decomposition is

x 1 2 x 1

x 1

x 1 2

x 1

Multiplying by the least common denominator, x

1 2 x 1 , we get

4x A x 1 x 1 B x 1 C x 1 2

A C x2 B 2C x A B C

Now we equate coefﬁcients:

A B C

A B 2C

A B C

Solving, we obtain A 1, B 2, and C 1, so

x4 2x2 4x 1

dx y x 1

x3 x2 x 1

x 1

x 1 2

x 1

x ln x 1

x 1 K

x 1

478 |||| CHAPTER 7 TECHNIQUES OF INTEGRATION

CASE III N Q(x) contains irreducible quadratic factors, none of which is repeated.

If Q x has the factor ax2 bx c, where b2 4ac 0, then, in addition to the partial fractions in Equations 2 and 7, the expression for R x Q x will have a term of the form

Ax B

ax2 bx c

where A and B are constants to be determined. For instance,				the function given by
f x x x 2 x2 1 x2 4 has a partial fraction decomposition of the form
	x	A	Bx C	Dx E

	x 2 x2 1 x2 4	x 2	x2 1	x2 4

The term given in (9) can be integrated by completing the square and using the formula

tan 1

x2 a2

EXAMPLE 5 Evaluate y

2x2 x 4

dx.

x3 4x

SOLUTION Since x3 4x x x2 4 can’t be factored further, we write

2x2 x 4

Bx C

x x2 4

x2 4

Multiplying by x x2 4 , we have

2x2 x 4 A x2 4 Bx C x

A B x2 Cx 4A

Equating coefﬁcients, we obtain

A B 2

C 1

4A 4

Thus A 1, B 1, and C 1 and so

2x2 x 4

x 1

x3 4x

x2 4

In order to integrate the second term we split it into two parts:

x 1

dx y

x2 4

We make the substitution u x2 4 in the ﬁrst of these integrals so that du 2x dx.

We evaluate the second integral by means of Formula 10 with a 2:

2x2 x 4

dx y

x x2 4

x2 4

ln x 21 ln x2 4 21 tan 1 x 2 K

SECTION 7.4 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS |||| 479

4x2 3x 2 EXAMPLE 6 Evaluate y 4x2 4x 3 dx.

SOLUTION Since the degree of the numerator is not less than the degree of the denominator, we ﬁrst divide and obtain

4x2 3x 2	1	x 1
4x2 4x 3	1	4x2 4x 3

Notice that the quadratic 4x2 4x 3 is irreducible because its discriminant is

b2 4ac 32 0. This means it can’t be factored, so we don’t need to use the partial fraction technique.

To integrate the given function we complete the square in the denominator:

4x2 4x 3 2x 1 2 2

This suggests that we make the substitution u 2x 1. Then, du 2 dx and x 12 u 1 , so

4x2 3x 2

x 1

dx y 1

4x2 4x 3

21 u 1 1

u 1

x 2

du x

4 y

u2 2

x 41 y

du 41 y

u2 2

x 81

tan 1

ln u2 2

x 81

ln 4x2 4x 3

tan 1

2x 1

C M

4s2

NOTE Example 6 illustrates the general procedure for integrating a partial fraction of

the form

Ax B
		where b2 4ac 0
ax2
	bx c

We complete the square in the denominator and then make a substitution that brings the integral into the form

Cu D

du C y

du D y

u2 a2

Then the ﬁrst integral is a logarithm and the second is expressed in terms of tan 1.

CASE IV

N Q(x) contains a repeated irreducible quadratic factor.

If Q x has the factor ax2 bx c r, where b2 4ac

0, then instead of the single

partial fraction (9), the sum

A1 x B1

A2 x B2

Ar x Br

ax2 bx c

ax2 bx c 2

ax2 bx c r

480 |||| CHAPTER 7 TECHNIQUES OF INTEGRATION

N It would be extremely tedious to work out by hand the numerical values of the coefﬁcients in Example 7. Most computer algebra systems, however, can ﬁnd the numerical values very quickly. For instance, the Maple command

convert f, parfrac, x

or the Mathematica command

	Apart[f]
gives the following values:
A 1,	B 81 , C D 1,
E 158 ,	F 81 , G H 43 ,
	I 21 , J 21

N In the second and fourth terms we made the mental substitution u x 2 1.

occurs in the partial fraction decomposition of R x Q x . Each of the terms in (11) can be integrated by ﬁrst completing the square.

EXAMPLE 7 Write out the form of the partial fraction decomposition of the function

x3 x2 1

x x 1 x2 x 1 x2 1 3

SOLUTION

x3 x2 1

x x 1 x2 x 1 x2 1 3

Cx D

Ex F

Gx H

Ix J

x 1

x2 x 1

x2 1

x2 1 2

x2 1 3

EXAMPLE 8 Evaluate y

1 x 2x2 x3

dx.

x x2 1 2

SOLUTION The form of the partial fraction decomposition is

1 x 2x2 x3

Bx C

Dx E

x x2 1 2

x2 1

x2 1 2

Multiplying by x x2 1 2, we have

x3 2x2 x 1 A x2 1 2 Bx C x x2 1 Dx E x

A x4 2x2 1 B x4 x2 C x3 x Dx2 Ex

A B x4 Cx3 2A B D x2 C E x A

If we equate coefﬁcients, we get the system

A B 0

C 1

2A B D 2

C E 1

A 1

which has the solution A 1, B 1, C 1, D 1, and E 0. Thus

1 x 2x2 x3

x 1

dx y

x x2 1 2

x2 1

x2 1 2

x dx

x2 1

x2 1 2

ln x 21 ln x2 1

tan 1x

2 x2 1

We note that sometimes partial fractions can be avoided when integrating a rational function. For instance, although the integral

x2 1

y x x2 3 dx

SECTION 7.4 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS |||| 481

could be evaluated by the method of Case III,			it’s much easier to observe that if
u x x2 3 x3 3x, then du 3x2 3 dx and so
y	x2 1
		dx 31 ln x	3 3x C
	x x2 3

RATIONALIZING SUBSTITUTIONS

Some nonrational functions can be changed into rational functions by means of appropriate substitutions. In particular, when an integrand contains an expression of the form sn t x , then the substitution u sn t x may be effective. Other instances appear in the exercises.

EXAMPLE 9 Evaluate y sx x 4 dx.

SOLUTION Let u sx 4 . Then u2 x 4, so x u2 4 and dx 2u du.

Therefore

sx 4

dx y

2u du 2 y

u2 4

2 y 1

u2 4

We can evaluate this integral either by factoring u2 4 as u 2 u 2 and using

partial fractions or by using Formula 6 with a 2:

dx 2 ydu 8 y

x 4

u2 4

2u 8

u 2

2 ln

x 4

sx 4 2

7.4EXERCISES

1–6 Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefﬁcients.

(a)

(b)

x 3 3x 1

x 3 2x 2 x

(a)

(b)

x 2

x 2 x 2

(a)

x 4 1

(b)

x 5

4x 3

x 2 9 2

(a)

x 3

(b)

2x 1

x 2

4x 3

x 1 3 x 2 4 2

(a)

x 4

(b)

t 4 t 2 1

x 4 1

t 2 1 t 2 4 2

(a)

x 4

(b)

x 3 x x 2 x 3

x 6 x 3

7–38 Evaluate the integral.

x 6

x 9

10.

x 5 x 2

t 4 t 1

482

||||

CHAPTER 7

TECHNIQUES OF INTEGRATION

x 1

11.

12.

x 2 1

x 2 3x 2

13.

14.

x 2 bx

x a x b

4 x 3 2x 2 4

1 x 3 4x 10

15.

16.

x 3 2x 2

x 2 x 6

2 4y 2 7y 12

x 2 2x 1

17.

18.

y y 2 y 3

x 3 x

19.

20.

x 2 5x 16

x 5 2 x 1

2x 1 x 2 2

21.

22.

x 2

s 2 s 1 2

23.

5x 2 3x 2

24.

x 2 x 6

x 3 2x 2

x 3 3x

25.

26.

x 2 x 1

x 1 x 2 9

x 2 1 2

27.

x 3 x 2 2x 1

28.

x 2 2x 1

x 2 1 x 2 2

x 1 2 x 2 1

29.

x 4

30.

3x 2 x 4

x 2 2x 5

x 4 3x 2 2

31.

32.

x 3 1

x 2 4x 13

x 3 2x

x 3

33.

34.

x 4 4x 2 3

x 3 1

35.

36.

x 4 3x 2 1

x x 2 4 2

x 5 5x 3 5x

37.

x 2 3x 7

38.

x 3 2x 2 3x 2

x 2 4x 6 2

x 2 2x 2 2

39–50 Make a substitution to express the integrand as a rational function and then evaluate the integral.

39.

40.

x s

2 s

x 1

x 3

41.

42.

x 4

x 3

43.

44.

y1 3

x 2 x

x 2 1

45.

[Hint: Substitute u sx .]

sx sx

1 s

46.

e 2x

47. y e 2x 3e x 2 dx

cos x

48. y sin2x sin x dx

sec 2 t

49. y tan2 t 3 tan t 2 dt

e x

50. y e x 2 e 2x 1 dx

51–52 Use integration by parts, together with the techniques of this section, to evaluate the integral.

	51.	yln x 2 x 2 dx	52. yx tan 1x dx
;53.		Use a graph of f x 1 x 2 2x 3 to decide whether

x02 f x dx is positive or negative. Use the graph to give a rough estimate of the value of the integral and then use partial fractions to ﬁnd the exact value.

;54. Graph both y 1 x 3 2x 2 and an antiderivative on the same screen.

55–56 Evaluate the integral by completing the square and using Formula 6.

55. y	dx	56. y	2x 1
				dx
	x 2 2x		4x 2 12x 7

57.The German mathematician Karl Weierstrass (1815–1897) noticed that the substitution t tan x 2 will convert any rational function of sin x and cos x into an ordinary rational function of t.

(a) If t tan x 2 , x , sketch a right triangle or use

trigonometric identities to show that

cos

and

sin

s1 t 2

t 2

(b) Show that

cos x

1 t 2

and

sin x

1 t 2

58 –61 Use the substitution in Exercise 57 to transform the integrand into a rational function of t and then evaluate the integral.

58.	y	dx
		3 5 sin x
		1		2	1
59.	y		dx	60. y 3		dx
		3 sin x 4 cos x			1 sin x cos x

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