Пример 1. Вычислить 1. Разложение дроби в сумму простейших

Ïðèìåð 1.
Âû÷èñëèòü
Z
x
dx
x − 1)(x + 2)(x − 3)
(
1. Ðàçëîæåíèå äðîáè â ñóììó ïðîñòåéøèõ äðîáåé èùåì â âèäå
x
(
x − 1)(x + 2)(x − 3)
=
A
+
B
C
+
x−1 x+2 x−3
2. Ïðèâîäÿ ê îáùåìó çíàìåíàòåëþ ïðàâóþ ÷àñòü, èìååì
(
x
A(x + 2)(x − 3) + B(x − 1)(x − 3) + C (x − 1)(x + 2)
=
x − 1)(x + 2)(x − 3)
(x − 1)(x + 2)(x − 3)
3. Ïðèðàâíèâàÿ ÷èñëèòåëè äðîáåé, ïîëó÷àåì òîæäåñòâî
x = A(x + 2)(x − 3) + B(x − 1)(x − 3) + C (x − 1)(x + 2)
4. Ïîäñòàâëÿÿ â òîæäåñòâå âìåñòî x äåéñòâèòåëüíûå êîðíè çíàìåíàòåëÿ äàííîé ðàöèîíàëüíîé äðîáè, íàõîäèì A, B è C .
Ïîëàãàÿ â òîæäåñòâå x = 1, èìååì
1=
A(1 + 2)(1 − 3) + B(1 − 1)(1 − 3) + C (1 − 1)(1 + 2),
îòêóäà 1 = −6A è A = −1/6.
Ïîëàãàÿ x = −2, èìååì
−2 = A(−2 + 2)(−2 − 3) + B(−2 − 1)(−2 − 3) + C (−2 − 1)(−2 + 2),
îòêóäà −2 = 15B, B = −2/15.
Ïîëàãàÿ x = 3, èìååì
3=
A(3 + 2)(3 − 3) + B(3 − 1)(3 − 3) + C (3 − 1)(3 + 2),
îòêóäà 3 = 10C , C = 3/10.
Z
x
dx =
5. Ñëåäîâàòåëüíî,
(x − 1)(x + 2)(x − 3)
Z
−1/6 −2/15 3/10
=
+
+
dx =
x−1 x+2 x−3
Z
Z
Z
dx
dx
dx
= −1/6
− 2/15
+ 3/10
x−1
x+2
x−3
=
=
−1/6 ln |x − 1| − 2/15 ln |x + 2| + 3/10 ln |x − 3| + const.
1
Ïðèìåð 2.
Âû÷èñëèòü
Z
x−1
dx
(x2 + 1)(x2 + 2)
1. Ðàçëîæåíèå äðîáè â ñóììó ïðîñòåéøèõ äðîáåé èùåì â âèäå
(
x−1
Ax + B Cx + D
=
+
x2 + 1)(x2 + 2) x2 + 1 x2 + 2
2. Ïðèâîäÿ ê îáùåìó çíàìåíàòåëþ ïðàâóþ ÷àñòü, èìååì
x−1
(x2 + 1)(x2 + 2)
=
(
Ax + B)(x2 + 2) + (Cx + D)(x2 + 1)
(x2 + 1)(x2 + 2)
3. Ïðèðàâíèâàÿ ÷èñëèòåëè äðîáåé, ïîëó÷àåì òîæäåñòâî
x − 1 = (Ax + B)(x2 + 2) + (Cx + D)(x2 + 1).
Ïåðåïèøåì åãî â âèäå
x − 1 = Ax3 + 2Ax + Bx2 + 2B + Cx3 + Cx + Dx2 + D
èëè
x − 1 = (A + C )x3 + (B + D)x2 + (2A + C )x + (2B + D).
Ïðèðàâíèâàÿ êîýôôèöèåíòû ïðè îäèíàêîâûõ ñòåïåíÿõ x, ïîëó÷àåì ñèñòåìó óðàâíåíèé
A +C
B+D

2A + C




2B + D







=
0,
=
0,
=
1,
=
−1.
îòêóäà


A +C
2A + C


B+D
2B + D
=
0,
=
1.
=
0,
=
−1.


A
−2C + C


B
−2D + D
=
−C ,

A
=
1,
=
1.
C
=
−1.
=
−D,

B
=
−1,
=
−1.
D
=
1.
2
Z
4. Ñëåäîâàòåëüíî,
(
Z
x−1
dx =
x2 + 1)(x2 + 2)
x − 1 −x + 1
=
+
dx =
x2 + 1 x2 + 2
Z
x
−1
−x
1
dx =
=
+
+
+
x2 + 1 x2 + 1 x2 + 2 x2 + 2
Z
Z
Z
Z
xdx
dx
xdx
dx
=
− 2
− 2
+
=
x2 + 1
x +1
x + 2 x2 + 2
Z
Z
1
dx2
1
dx2
1
x
=
−
arctg x −
+ √ arctg √ =
2
2
x +1
2
x2 + 2
2
2
1
1
1
x
2
2
ln(x + 1) − arctg x − ln(x + 2) + √ arctg √
=
2
Ïðèìåð 3.
2
Âû÷èñëèòü
Z
x
4
2
x+5
x
( + 1)2 ( + 2)
2
+ const.
dx
1. Ðàçëîæåíèå äðîáè â ñóììó ïðîñòåéøèõ äðîáåé èùåì â âèäå
4
x
x+5
x
( + 1)2 ( + 2)
=
A
B
C
+
+
x + 1 (x + 1)2 x + 2
2. Ïðèâîäÿ ê îáùåìó çíàìåíàòåëþ ïðàâóþ ÷àñòü, èìååì
x+5
A(x + 1)(x + 2) + B(x + 2) + C (x + 1)2
=
(x + 1)2 (x + 2)
(x + 1)2 (x + 2)
4
3. Ïðèðàâíèâàÿ ÷èñëèòåëè äðîáåé, ïîëó÷àåì òîæäåñòâî
4
x + 5 = A(x + 1)(x + 2) + B(x + 2) + C (x + 1)2
4. Ïîäñòàâëÿÿ â òîæäåñòâå âìåñòî x äåéñòâèòåëüíûå êîðíè çíàìåíàòåëÿ äàííîé ðàöèîíàëüíîé äðîáè, íàõîäèì B, C .
Ïîëàãàÿ â òîæäåñòâå x = −1, èìååì
4 (−1) + 5 =
A(−1 + 1)(−1 + 2) + B(−1 + 2) + C (−1 + 1)2 ,
îòêóäà B = 1.
Ïîëàãàÿ x = −2, èìååì
4 (−2) + 5 =
A(−2 + 1)(−2 + 2) + B(−2 + 2) + C (−2 + 1)2
3
îòêóäà −3 = C , C = −3.
Äàëåå, ïîëàãàÿ x = 0, èìååì
4·0+5 =
A(0 + 1)(0 + 2) + B(0 + 2) + C (0 + 1)2
îòêóäà 5 = 2A + 2B + C , 5 = 2A + 2 − 3 è A = 3.
Z
4x+5
dx =
5. Ñëåäîâàòåëüíî,
2
x
x
( + 1) ( + 2)
Z
=
Z
=3
3
1
+
+
dx =
x + 1 (x + 1)2 x + 2
Z
Z
dx
dx
dx
+
−3
=
x + 1 (x + 1)2
x+2
x
= 3 ln | + 1| −
Ïðèìåð 4.
−3
1
x+1
− 3 ln |x + 2| + const.
Âû÷èñëèòü
Z
(
3x+3
dx
x − 1) (x2 + 2)
1. Ðàçëîæåíèå äðîáè â ñóììó ïðîñòåéøèõ äðîáåé èùåì â âèäå
(
3x+3
A Bx + C
=
+
x − 1) (x2 + 2) x − 1 x2 + 2
2. Ïðèâîäÿ ê îáùåìó çíàìåíàòåëþ ïðàâóþ ÷àñòü, èìååì
x+3
A(x2 + 2) + (Bx + C )(x − 1)
=
2
(x − 1) (x + 2)
(x − 1) (x2 + 2)
3
3. Ïðèðàâíèâàÿ ÷èñëèòåëè äðîáåé, ïîëó÷àåì òîæäåñòâî
3
x + 3 = A(x2 + 2) + (Bx + C )(x − 1)
Ïîëàãàÿ â òîæäåñòâå x = 1, èìååì 6 = A · 3, îòêóäà A = 2.
Ïîëàãàÿ x = 0, èìååì 3 = 2A − C îòêóäà 3 = 4 − C è C = 1.
Ïîëàãàÿ x = −1, èìååì
0=3
A − 2(−B + C )
îòêóäà 0 = 6 − 2(−B + 1), 0 = 4 + 2B è B = −2.
4
4. Ñëåäîâàòåëüíî,
Z
Z
x+3
2
−2x + 1
dx =
dx =
+
(x − 1) (x2 + 2)
x − 1 x2 + 2
Z
Z
Z
Z
1
x
dx
2xdx
dx
dx2
=2
− 2
+
= 2 ln |x − 1| −
+ √ arctg √ =
x−1
x + 2 x2 + 2
x2 + 2
2
2
1
x
2
= 2 ln |x − 1| − ln(x + 2) + √ arctg √ + C
3
2
Ïðèìåð 5.
Z
Âû÷èñëèòü
Z
dx
1 + cos x + sin x
t = tg x
2
=
2 dt
dx =
2
dx
x + sin x
Z
1 + cos
1+
sin
x=
t
1 + t2
2
Z
dt
=
Ïðèìåð 6.
Íàéòè
1+
s
Z 4
t
x+2
x−1
= ln |1 +
3
t 2 1 + t2 =
1−
x=
Z
t
dt
1 + t2
2
1−t
2t
1+
+
2
1+t
1 + t2
cos
dt
1 + t2
2
=
2
2
=
1+
=
t 2 + 1 − t 2 + 2t
1 + t2
x
t | + const = ln |1 + tg | + const.
2
dx
x
x+2 4
Ïîäñòàíîâêà
= t ïðèâåäåò ê èíòåãðèðîâàíèþ ðàöèîíàëüíîé ôóíêöèè.
x−1
Èç óêàçàííîé ïîäñòàíîâêè îïðåäåëèì x, x + 2 à ïîòîì dx:
( + 2)2
x + 2 = t 4 (x − 1), x + 2 = xt 4 − t 4 ,
xt 4 − x = t 4 + 2, x(t 4 − 1) = t 4 + 2, x =
t4 + 2
;
t4 − 1
4
3t
t4 + 2
+2 =
.
t4 − 1
t4 − 1
3 4
4
3
4t (t − 1) − (t + 2)4t
−12t 3
dx =
dt , dx = 4
dt
4
2
(t − 1)
(t − 1)2
x+2 =
Ïîýòîìó
s
3
Z 4
x+2
x−1
dx
(x + 2)2
Z
=
(t
t3
4 − 1)2
9t 8
−12t 3
4
dt = −
4
2
(t − 1)
3
Z
dt
t2
=
41
3
t
+
C=
r
4 4
3
x−1
+C
x+2
5
√
Z √
x+3 4 x
√
√
Âû÷èñëèòü
3
6
Ïðèìåð 7.
dx
x + 2 x)x
(
Òàê êàê ÍÎÊ(2, 3, 4, 6) = 12. Ïðèìåíèì ïîäñòàíîâêó x = t 12 , òîãäà dx = 12t 11 dt
è, ñëåäîâàòåëüíî,
√
Z √
x+3 4 x
√
√
3
6
Z
Z
3
3
t 6 + 3t 3
(t + 3)t
· 12t 11 dt = 12
dt =
4
2
12
2
2
(t + 2t )t
(t + 2)t t
( x + 2 x)x
Z 3
Z
Z
t +3
−2t + 3
2t
3
= 12
dt
= 12
t
+
dt
= 12
t
−
+
dt =
t2 + 2
t2 + 2
t2 + 2 t2 + 2
2
√
√ 6
12
√
t
3
x
x
t
3
= 12
− ln(t 2 + 2) + √ arctg √ + C = 12
− ln( 6 x + 2) + √ arctg √
+C,
dx =
2
2
ïîñêîëüêó t =
2
2
2
2
√
12
x.
Z
x2 dx
.
(3 − x2 )3/2
√
Ïðèìåíèì ïîäñòàíîâêó x = 3 sin t , îòêóäà
Ïðèìåð 8.
Âû÷èñëèòü
(3 −
x2 )3/2 = (3 − 3 sin2 t )3/2 = 3
dx =
√
3 cos
√
3 cos
3 t,
t dt .
Òîãäà ïîëó÷èì
Z
x2 dx
(3 − x2 )3/2
Z
=
3 sin
√
3
2t
3 cos3
t
·
√
3 cos
t dt =
Z
tg
2 t dt
Z
−
1 dt = tg t − t + C
cos2 t
1
=
Òàê êàê
x
t = arcsin √ ,
3
òî
Z
t
1−
cos =
x2 dx
(3 − x2 )3/2
Z
r
=
√
x
x2
3
,
t √
tg =
x
− arcsin √
3 − x2
3
x
3−
+
x2
,
C.
dx
(3 + x2 )3/2
√
Ïðèìåíèì ïîäñòàíîâêó x = 3 tg t , îòêóäà
Ïðèìåð 9.
Âû÷èñëèòü
(3 +
x2 )3/2 = (3 + 3 tg2 t )3/2 =
3
cos2
√
3/2
t
=
3
3
cos3
t
,
√
dx =
3
cos2
t
dt .
6
Òîãäà ïîëó÷èì
Z
Z
dx
(3 + x2 )3/2
3
√ ·
=
3
Òàê êàê
cos2 t
3
dt =
1
cos
3
3
x
1
,
= 1+
2
sin t
x
t √
sin =
òî
Z
dx
x2 )3/2
(3 +
Ïðèìåð 10.
Âû÷èñëèòü
Z√ 2
x −1
x4
Ïðèìåíèì ïîäñòàíîâêó x =
1
cos
t
x2 − 1 =
p
dx =
x4
Z
4t
t
tg cos
sin
t
cos2 t
=
1
√
3
x2
=
3+
x2
x2
3
t C.
sin +
,
,
x
3+
x2
+
C.
dx.
, îòêóäà
r
dx =
Òîãäà ïîëó÷èì
Z√ 2
x −1
x2
3+
3
1
t dt =
√
t
ctg =
dt =
1
cos2
sin
t
t
Z
sin
− 1 = tg t ,
dt .
cos2 t
2 t cos t dt
Òàê êàê
Z
=
2 t d sin t
sin
=
3t
sin
3
+
C.
√
t
cos =
òî
1
x
Z√ 2
x −1
x4
Z
Ïðèìåð 11.
Z
Z
√
3t
cos
Íàéòè
ln
,
x2 − 1
,
x
t
sin =
dx =
x2 − 1)3/2
+C
x3
1(
3
xdx
Z
Z
1
Z
x d |{z}
x = x ln x − xd ln x = x ln x − x · dx = x ln x − dx = x ln x − x + C .
|{z}
x
ln
u
v
Ïðèìåð 12.
Z
Z
Íàéòè
Z
x cos xdx
x cos xdx = xd sin x = x sin x −
Z
sin
xdx = x sin x + cos x + C .
7
Z
Ïðèìåð 13.
Z
Íàéòè
u = ln x
3
4x ln xdx = dv = 4x3 dx
x3 ln xdx
4
1
v = x4
4
= x ln x −
Ïðèìåð 14.
du = dx
x =
Z
x4 ·
1
dx = x4 ln x −
x
Z
1
x3 dx = x4 ln x − x4 + C .
4
Âû÷èñëèòü ïëîùàäü ôèãóðû, îãðàíè÷åííîé ãðàôèêàìè ôóíê-
öèé.
x2 = 4y, y =
8
4+
x2
.
y
−2
S=2
Z2 0
8
−
4 + x2
Ïðèìåð 15.
x2
4
dx = 2
4 arctg
O
x
2
−
x
2
x3 2
12
0
=2
4 arctg 1 −
8
12
=2
p−
2
3
Âû÷èñëèòü ïëîùàäü ôèãóðû, îãðàíè÷åííîé ëèíèÿìè, çàäàííû-
ìè óðàâíåíèÿìè.

x = 3 cos t ,
 y = 2 sin t ,
0
≤ t ≤ 2p
8
y
2
t = p /2
t =0
x
3
O
−3
t
−2
Z3
S = 4 ydx = 4
0
Z p /2
= 12
 Ïðèìåð
x = cos t ,
16.
0
Z0
p /2
t
0
2 sin 3 cos
p
0
x
3
y
0
t dt = 24
p
6
√
3
3
4
√
3
2
√2
1
Z p /2
2
2 t dt
sin
0
2
p
p
3
3
2
√2
3
0
2
=
p /2
(1 − cos 2t ) dt = 12 (t − sin 2t )
= 6p .
2
1
0
Âû÷èñëèòü
ïëîùàäü
ôèãóðû,
îãðàíè÷åííîé
ëèíèÿìè
y = 0.
y = 2 sin2 t ,
y = 2 sin2 t
1
t
p
O
2
x = cos t
x2 = cos2 t ,
y
y
2
−1
O
1
y
x
S=2
2
2 t,
= sin
+
y = −2x2 + 2.
Z0
p /2
2 sin
=4
Ïðèìåð 17.
x2 = 1,
cos
2 t d (cos t ) = 4
t−
Z0
3 t 0
cos
3
p /2
p /2
2 t ) d (cos t ) =
(1 − cos
=4
1−
1
3
=4
2
3
.
Âû÷èñëèòü ïëîùàäü ïëîñêîé ôèãóðû, îãðàíè÷åííîé îäíîé àð-
êîé öèêëîèäû x = t − sin t , y = 1 − cos t è îñüþ Ox.
9
t0 = 0 , t 1 = 2 p ,
dx = (1 − cos t ) dt
y
2
p
O
p x
2
S=
Z 2p
0
Z 2p =
0
t 2 dt =
(1 − cos )
t
1 − 2 cos +
t
1 + cos 2
2
0
dt =
1
1
2
4
t − 2 sin t + t +
=
Ïðèìåð 18.
Z 2p
2 t ) dt
t
(1 − 2 cos + cos
Z 2p t
1 − 2 cos +
0
2p
t sin 2
0
=2
1
2
+
=
t
cos 2
2
dt =
p + p = 3p
Âû÷èñëèòü ïëîùàäü ôèãóðû, îãðàíè÷åííîé ëèíèÿìè, çàäàííû-
ìè â ïîëÿðíûõ êîîðäèíàòàõ.
r2 = cos 2j
r=
1
O
p
4
√
cos 2
j
O
p
j
r = cos 2j r = cos j
S=4
1
2
Z p /4
0
2p
3
1
−1
O
cos 2
P
j d j = sin 2j |0p /4 = 1.
p
6
1
−1
Ïðèìåð 19.
10
r = sin 3j
D = {(r, j ) : 0 ≤ j ≤ p /3, 0 ≤ r ≤ sin 3j }
S = 3SD =
=
3
4
Z p /3
0
3
2
Z p /3
0
2
sin 3
j dj =
j) dj =
(1 − cos 6
Ïðèìåð 20.
p
4
Âû÷èñëèòü äëèíó äóãè êðèâîé, çàäàííîé óðàâíåíèÿìè â ïðÿ-
ìîóãîëüíîé ñèñòåìå êîîðäèíàò.
y=
p
1−
x2 + arcsin x,
L=
≤ x ≤ 7/9
y x dx
1 + ( 0 ( ))2
a
x
1−x
y=
+√
= √
=
2
2
1+x
1−x
1−x
1−x
2
0
2
1 + (y (x)) = 1 +
=
1+x
1+x
√
7/9
√ Z 7/9 dx
√ √
√ 4
2 2
√
L= 2
=2 2
1 + x
=2 2
−1 =
3
3
0
1+x
0
0
Ïðèìåð 21.
óðàâíåíèÿìè.
−2x
√
2 1 − x2
Zb p
0
1−
1
r
Âû÷èñëèòü äëèíó äóãè êðèâîé, çàäàííîé ïàðàìåòðè÷åñêèìè

x = 4 sin t + 3 cos t ,
y = 3 sin t − 4 cos t ,
L=
Z t1 q
t0
0
≤ t ≤ p /2.
xt0 )2 + (yt0 )2 dt
(
xt0 = 4 cos t − 3 sin t ,
yt0 = 3 cos t + 4 sin t ,
xt0 )2 = (4 cos t − 3 sin t )2 = 16 cos2 t − 24 cos t sin t + 9 sin2 t ,
2
0 2
2
2
(yt ) = (3 cos t + 4 sin t ) = 9 cos t + 24 cos t sin t + 16 sin t ,
(
11
(
xt0 )2 + (yt0 )2 = 25 cos2 t + 25 sin2 t = 25
L=
Z p /2
5
0
p
5
dt =
2
Âû÷èñëèòü äëèíó äóãè êðèâîé, çàäàííîé óðàâíåíèÿìè â ïîëÿð-
Ïðèìåð 22.
íûõ êîîðäèíàòàõ.
r = 1 + cos j ,
L=
Ïðèìåð
23.
r2 + (rj0 )2 d j
a
r0 = − sin j
Zp p
0
≤j ≤p
Zb q
L=
Zp r
0
(1 + cos j )2 + (− sin j )2 d j
0
2+2
2 cos2
j
2
− 1 dj = 2
=
Zp p
Zp
0
cos
0
j
2
2 + 2 cos
j dj =
j p
d j = 4 sin = 4
2
0
Âû÷èñëèòü îáúåì òåëà, îáðàçîâàííîãî âðàùåíèåì ôèãóðû,
îãðàíè÷åííîé ãðàôèêàìè ôóíêöèé. Îñü âðàùåíèÿ Ox.
y = sin(p x/2), y = x2 .
y
O
V = V1 − V2 = p
=
p
2
Z1
0
(1 − cos
Ïðèìåð
24.
p x) dx − p
Z1
0
2
sin
x5 1 p
5
0
=
1
2
px
2
x−
x
dx − p
1
p
Z1
0
x4 dx =
1
p
sin p x −
5
0
=
p
2
−
p
5
=
p
3
10
Âû÷èñëèòü îáúåì òåëà, îáðàçîâàííîãî âðàùåíèåì ôèãóðû,
îãðàíè÷åííîé ãðàôèêàìè ôóíêöèé. Îñü âðàùåíèÿ Oy.
y = arcsin x, y = arccos x, y = 0
12
y
x
x
O
p
O
y
4
V = V1 − V2 = p
=
p
Z p /4 y
1 + cos 2
2
0
Ïðèìåð 25.
Z +∞
0
−∞
Ïðèìåð 27.
1
Z −1
−2
Z p /4
2 ydy =
sin
0
x
0
1
1
0
2
Z0
arctg 2
−∞
2
x
x→−∞
=
1
0−
4
Ze
=
1
d ln x
2
1 − ln x
p
x
xdx 1
=
x2 − 1 2
−2
dx2
x2 − 1
=
arctg 2
x
1 + 4x2
2
−
=
Ze
1
=
p
2
=
2
dx
0
x−∞ =
p2
16
dx
x
e
p
1 − ln
2x
e
2
−2
Íàéòè ñóììó ðÿäà
1
1·4
1
+
n(n + 3)
1
2·5
=
lim
Z −1
−2
1
x→−1−0 2
+
1
3·6
.
1
= arcsin(ln )|1 = arcsin(ln ) − arcsin(ln 1) =
−1
1
2
ln |x − 1|
=
an =
2
arctg 2
p2
Âû÷èñëèòü íåñîáñòâåííûé èíòåãðàë
Z −1
0
e−x − 1
x→+∞
lim
−∞
4
p / 4
y
2
Z0
1
4
2
sin 2
xe−x dx
0
2
xd arctg 2x =
0 − lim arctg 2
4
Z +∞
2
2 +∞
1
1
e−x dx2 = − e−x = −
dx =
p
ydy =
cos 2
Âû÷èñëèòü íåñîáñòâåííûé èíòåãðàë
2
1 − ln x
Ïðèìåð 29.
dy = p
2
0
1 + 4x2
p
Ïðèìåð 28.
2
Z +∞
2
arctg 2
dx
x
y
1 − cos 2
−
Z p /4
Âû÷èñëèòü íåñîáñòâåííûé èíòåãðàë
=
Ze
1
2
Z0
0
2 ydy − p
cos
Âû÷èñëèòü íåñîáñòâåííûé èíòåãðàë
xe−x dx =
Ïðèìåð 26.
Z p /4
p
2
xdx
x2 − 1
x2 − 1| − ln 3) = ∞
(ln |
+...+
1
n(n + 3)
+...
A B
+
n n+3
13
1
A(n + 3) + Bn
=
n(n + 3)
n(n + 3)
1 = A(n + 3) + Bn
n = 0,
1=3
n = −3,
an =
sn =
1
3
+
1
3
1−
1
6
−
1
+
4
1
+
9
1
3
1
1
2
3
1
3
1−
+
1
4
1
n−3
+
−
−
1
1
1
1
+
+
1
1+
3
1
Ïðèìåð 30.
11
18
−
6
+
1
4
−
1
1
7
+
1
+
1
−
3
1
n+1
1
+
2
1
1+
3
2
−
1
1
1
5
=
1
1
1
−
+
n
1
8
+
1
1
1
6
1
−
1
1
9
+
1
7
1
−
n+3
1
1
3
=
−
8
1
+
n+1
+
=
+
−
n−2
3
n+2 n
n+2
1
−
1
n+3
1
5
1
−
+...
10
n+3
=
.
1
1
−
−
−
3
n+1 n+2 n+3
+
1
−
+
7
3
1
−
−
1
4
−
n
3
n(n + 3)
3
n−3
+
1
1
3
.
n+3
+...
1
1+
1
1
1
1
+
3
1
1
−
n+1 n−1
+
2
=
1
6
−
1
s = nlim
sn = nlim
→∞
→∞
Èòàê, ñóììà ðÿäà s =
3
3
n
3·6
n+2
−
3
1
1
1
1
−
1
B=− ,
1
3
+
+...+
n n−2
=
3
1
=
1
5
1
+
5
1
1
2·5
n−1
−
2
1
+
−
10
3
=
1
1·4
7
1
n(n + 3)
3
−3B,
1=
1
1
A, A = ,
=
11
18
.
Èññëåäîâàòü ñõîäèìîñòü ðÿäà
Ñðàâíèì ýòîò ðÿä ñ ðÿäîì
ñêîé ïðîãðåññèåé).
an
n→∞ bn
lim
å
n=1
∞
1
2n
å
n=1
∞
(ò. å. ñ áåñêîíå÷íî óáûâàþùåé ãåîìåòðè÷ån
= lim
1
.
2n − 1
2
n→∞ 2n − 1
= lim
n→∞
1
1−
1
= 1.
2n
14
å
n=1
∞
Òàê êàê ïðåäåë êîíå÷åí è îòëè÷åí îò íóëÿ è ðÿä
äàííûé ðÿä.
1
ñõîäèòñÿ, òî ñõîäèòñÿ è
2n
Èññëåäîâàòü íà ñõîäèìîñòü ðÿä
Ïðèìåð 31.
å
n=1 n
∞
1
!
1
= 1+
2!
+
1
+...+
3!
1
n!
+...
Ïðèìåíèì ïðèçíàê Äàëàìáåðà:
an =
an+1
lim
n→∞ an
1
n!
an+1 =
,
n
1
( + 1)!
= lim
n→∞
1
n
( + 1)!
1
n!
= lim
=
n→∞ (n + 1)!
n!
n!
1
= lim
= lim
= 0 < 1.
n→∞ n!(n + 1) n→∞ n + 1
Äàííûé ðÿä ñõîäèòñÿ, òàê êàê ïðåäåë ìåíüøå åäèíèöû.
Ïðèìåð 32.
å
∞
n=1
Èññëåäîâàòü íà ñõîäèìîñòü ðÿä
n n 2 1 2 4 3 9
=
+
+
+...+
n+1
2
3
4
n n 2
n+1
+...
Ïðèìåíèì ïðèçíàê Êîøè:
lim
n→∞
√
n
an = nlim
→∞
n→∞
n
 1 
 n+1 
n n2
n+1
n

= lim
r
= lim
n→∞
n
= lim
n+1
n→∞
1
1+
1
n n
n
=
n
1
e
=
< 1.
Äàííûé ðÿä ñõîäèòñÿ, òàê êàê ïðåäåë ìåíüøå åäèíèöû.
Ïðèìåð 33.
Îïðåäåëèòü îáëàñòü ñõîäèìîñòè ñòåïåííîãî ðÿäà
1−
x
22
+
x2
32
x4
− 2
4
n−1 x
+ . . . + (−1)
an = lim
R = nlim
→∞ an+1 n→∞
n−1
n2
1
n2
n
1
( + 1)2
= lim
n→∞
n
+...
2
( + 1)
n2
= 1.
15
Òàêèì îáðàçîì, R = 1 è ðÿä ñõîäèòñÿ â èíòåðâàëå (−1, 1).
Åñëè x = −1, ïîëó÷àåì ÷èñëîâîé ðÿä
1+
1
22
+
1
32
+
1
42
+...+
1
Ðÿä ñ îáùèì ÷ëåíîì an = a ñõîäèòñÿ ïðè
n
â òî÷êå x = −1.
1
n2
+...
a > 1. Ñëåäîâàòåëüíî, ðÿä ñõîäèòñÿ
Ïðè x = 1 ïîëó÷àåì ðÿä Ëåéáíèöà
1−
1
>
1
4
>
1
9
>
1
22
1
+
1
1
32
− 2
4
n−1
+ . . . + (−1)
1
16
> ... > 2 >
n
n
1
( + 1)2
1
n2
+...
> ...,
lim
1
n→∞ n2
= 0.
Èòàê, äàííûé ðÿä ñõîäèòñÿ íà [−1, 1].
Ïðèìåð 34.
Ðåøèòü óðàâíåíèå x2 y2 y0 + 1 = y
Ïðèâîäèì óðàâíåíèå ê âèäó:
x2 y2
dy
+ 1 = y,
x2 y2 dy = (y − 1)dx.
dx
Äåëèì îáå ÷àñòè óðàâíåíèÿ íà x2 (y − 1):
y2
y−1
dy =
dx
x2
Ïåðåìåííûå ðàçäåëåíû. Èíòåãðèðóåì îáå ÷àñòè óðàâíåíèÿ:
Z
Z 2
Z
Z 2
Z
Z
y
dx
y −1+1
dx
1
dx
dy = 2 ,
=
,
y
+1+
=
,
y−1
x
y−1
x2
y−1
x2
y2
2
+
y + ln |y − 1| = −
1
x
+
C.
Ïðè äåëåíèè íà x2 (y − 1) ìîãëè áûòü ïîòåðÿíû ðåøåíèÿ
x = 0 è y − 1 = 0, ò. å.
y = 1. Î÷åâèäíî, y = 1 ðåøåíèå óðàâíåíèÿ, à x = 0 íåò.
Ïðèìåð 35.
Ðåøèòü óðàâíåíèå
x
Ïîëàãàåì y = uv, òîãäà
dy
− 2y = 2x4
dx
dy dv du
=u
+
v
dx dx dx
16
dy
â èñõîäíîå óðàâíåíèå, áóäåì èìåòü
dx
dv du
dv
du
x(u + v) − 2uv = 2x4 , xu − 2uv + x v = 2x4
dx dx
dx
dx
Ïîäñòàâëÿÿ âûðàæåíèå
u x
dv
− 2v
dx
Äëÿ îïðåäåëåíèÿ v ïîëó÷èì óðàâíåíèå x
xdv = 2vdx,
dv dx
=2
, èëè
v
x
+
x
du
v = 2x4
dx
dv
− 2v = 0 îòêóäà
dx
v| = 2 ln |x| + C1 , v = x2 .
ln |
Ïîäñòàâëÿÿ âûðàæåíèå ôóíêöèè v = x2 â óðàâíåíèå
dv
du
u x − 2v + x v = 2x4 ,
dx
dx
ïîëó÷àåì äëÿ îïðåäåëåíèÿ u óðàâíåíèå
x3
îòêóäà
du
du
4
= 2x ,
èëè
= 2 x,
dx
dx
u = x2 + C . Ñëåäîâàòåëüíî, îáùèé èíòåãðàë çàäàííîãî óðàâíåíèÿ áóäåò
èìåòü âèä
y = Cx2 + x4 .
y00 − y = 0. Õàðàêòåðèñòè÷åñêîå óðàâíåíèå
k 2 − 1 = 0 èìååò ðàçëè÷íûå êîðíè k1 = 1, k2 = −1. ×àñòíûå ðåøåíèÿ: y1 = ex ,
y2 = e−x . Òîãäà îáùåå ðåøåíèå: y = C1 ex + C2 e−x
Ïðèìåð 36.
Ïðèìåð 37.
Ðåøèòü óðàâíåíèå
Ðåøèòü óðàâíåíèå y00 − 2y + y = 0.
Ïèøåì õàðàêòåðèñòè÷åñêîå óðàâíåíèå k 2 − 2k + 1 = 0. Íàõîäèì åãî êîðíè:
k1 = k2 = 1. Îáùèì èíòåãðàëîì áóäåò y = (C1 + C2 x)ex .
Ïðèìåð 38.
Ðåøèòü óðàâíåíèå y00 + y = 0.
Íàïèøåì õàðàêòåðèñòè÷åñêîå óðàâíåíèå k 2 + 1 = 0. Íàõîäèì åãî êîðíè: k1 =
i, k2 = −i.
Îáùåå ðåøåíèå äàííîãî óðàâíåíèÿ: y = C1 cos x + C2 sin x, ãäå C1 , C2 ïðîèçâîëüíûå ïîñòîÿííûå.
17
Ïðèìåð 39.
Íàéòè îáùåå ðåøåíèå óðàâíåíèÿ ìåòîäîì âàðèàöèè ïðîèçâîëü-
íûõ ïîñòîÿííûõ:
y00 + y = tg x.
Õàðàêòåðèñòè÷åñêîå óðàâíåíèå k 2 + 1 = 0 èìååò êîðíè k1,2 = ±i. Îáùåå ðåøåíèå îäíîðîäíîãî óðàâíåíèÿ y = C1 cos x + C2 sin x.
Ðåøàÿ ñèñòåìó
y1 (x) = cos x, y2 (x) = sin x,

y (x)
 1
y01 (x)
 





y2 (x) C1   0 
·
=
,
y02 (x)
C20
f (x)
0
C10 cos x+C20 sin x =0,
−C 0 sin x+C 0 cos x=tg(x),
1
2
ïîëó÷àåì C10 = − sin2 x/ cos x, C20 = sin x, îòêóäà
Z
x p 2
cos x − 1
C1 =
dx = sin x − ln tg + + A1 ,
x
C2 = − cos x + A2 .
cos
2
4
Òàêèì îáðàçîì, îáùåå ðåøåíèå èñõîäíîãî óðàâíåíèÿ
x p y = sin x − ln tg + + A1 · cos x + (− cos x + A2 ) · sin x =
2
4
x p = A1 cos x + A2 sin x − ln tg
+
· cos x.
2
4
Ïóñòü ïðàâàÿ ÷àñòü ëèíåéíîãî íåîäíîðîäíîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ èìååò âèä:
f (x) = P(x)ea x
ãäå P(x) = pm xm + . . . + p0 ìíîãî÷ëåí ñòåïåíè m ≥ 0.
Òîãäà ÷àñòíîå ðåøåíèå èùåòñÿ â âèäå:
y(x) = xr Q(x)ea x .
Q(x) ìíîãî÷ëåí òîé æå ñòåïåíè, ÷òî è P(x), íî ñ íåîïðåäåë¼ííûìè êîýôôèöèåíòàìè, à r ≥ 0 êðàòíîñòü êîðíÿ a õàðàêòåðèñòè÷åñêîãî óðàâíåíèÿ.
Çäåñü
Ïðèìåð 40.
Ðåøèòü óðàâíåíèå y000 − y0 = x.
Ðåøèì ñîîòâåòñòâóþùåå îäíîðîäíîå óðàâíåíèå: y000 − y0 = 0.
k 3 − k = 0, k1 = 0, k2 = 1, k3 = −1,
y = C1 + C2 ex + C3 e−x .
18
Íàéäåì ÷àñòíîå ðåøåíèå èñõîäíîãî íåîäíîðîäíîãî óðàâíåíèÿ.
P(x)ea x = x = xe0x .
×àñòíîå ðåøåíèå èùåì â âèäå: y = xr Q(x)ea x , ãäå r = 1, a
y = Ax2 + Bx.
= 0,
Q(x) = Ax + B. Ò. å.
Îïðåäåëèì íåèçâåñòíûå êîýôôèöèåíòû A è B.
y0 = 2Ax + B, y00 = 2A, y000 = 0,
−2Ax − B = x,
−2A = 1,
A=−
1
2
;
B = 0.
x2
y∗ = − . Îáùåå ðåøåíèå ëèíåéíîãî íåîäíîðîäíîãî äèôôå2
x2
ðåíöèàëüíîãî óðàâíåíèÿ: y = C1 + C2 ex + C3 e−x − .
×àñòíîå ðåøåíèå:
2
Ïðèìåð 41.
Ðåøèòü óðàâíåíèå y
00 + 2 0 + 2 = 2 −2x .
y
y
e
Ðåøèì ñîîòâåòñòâóþùåå îäíîðîäíîå óðàâíåíèå: y00 + 2y0 + 2y = 0.
k 2 + 2k + 2 = 0, k1 = −1 − i, k2 = −1 + i,
y = e−x (C1 cos x + C2 sin x).
Íàéäåì ÷àñòíîå ðåøåíèå èñõîäíîãî íåîäíîðîäíîãî óðàâíåíèÿ.
P(x)ea x = 2e−2x .
y = xr Q(x)ea x , ãäå r = 0, a = −2, Q(x) = A. Ò. å.
y = Ae−2x . Îïðåäåëèì íåèçâåñòíûé êîýôôèöèåíò A.
×àñòíîå ðåøåíèå èùåì â âèäå:
y0 = −2Ae−2x , y00 = 4Ae−2x ,
4
Ae−2x − 4Ae−2x + 2Ae−2x = 2e−2x ,
A = 2, A = 1.
2
y∗ = e−2x . Îáùåå ðåøåíèå ëèíåéíîãî íåîäíîðîäíîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ: y = e−x (C1 cos x + C2 sin x) + e−2x .
×àñòíîå ðåøåíèå:
Ïðèìåð 42.
Ðåøèòü óðàâíåíèå y00 + 2y0 + y = (x − 6)e−x .
Ðåøèì ñîîòâåòñòâóþùåå îäíîðîäíîå óðàâíåíèå: y00 + 2y0 + y = 0.
k 2 + 2k + 1 = 0, k1 = −1, k2 = −1,
y = C1 e−x + C2 xe−x .
19
Íàéäåì ÷àñòíîå ðåøåíèå èñõîäíîãî íåîäíîðîäíîãî óðàâíåíèÿ.
P(x)ea x = (x − 6)e−x .
y = xr Q(x)ea x , ãäå r = 2, a = −1, Q(x) = Ax + B. Ò.
å. y = x2 (Ax + B)e−x . Îïðåäåëèì íåèçâåñòíûå êîýôôèöèåíòû A è B.
×àñòíîå ðåøåíèå èùåì â âèäå:
y00 + 2y0 + y = (6Ax + 2B)e−x ,
Ax + 2B)e−x = (x − 6)e−x ,
(6
6
A = 1, A =
1
6
;
2
B = −6, B = −3.
×àñòíîå ðåøåíèå: y∗ = 16 x3 − 3x2 e−x . Îáùåå ðåøåíèå ëèíåéíîãî íåîäíîðîäíî
ãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ: y = C1 e−x + C2 xe−x + 61 x3 − 3x2 e−x .
Ðàññòàâèòü
ïðåäåëû èíòåãðèðîâàíèÿ â òîì è äðóãîì ïîðÿäêå
ZZ
â äâîéíîì èíòåãðàëå f (x, y) dxdy, ãäå îáëàñòü D îãðàíè÷åíà ëèíèÿìè y = x2 ,
Ïðèìåð 43.
D
x + y = 2 , x = 0.
y
2
1
O
1
x
D ÿâëÿåòñÿ ñòàíäàðòíîé îòíîñèòåëüíî îáîèõ îñåé êîîðäèíàò:
D = {(x, y) | 0 ≤ x ≤ 1, x2 ≤ y ≤ 2 − x}.
ZZ
D
Z1
2Z−x
0
x2
f (x, y) dxdy = dx
f (x, y) dy.
20
x
y = x2
1
x+y = 2
O
1
y
2
Ïðèìåð 44.
Ïðåäñòàâèì ìíîæåñòâî D â âèäå D = D1 ∪ D2 , ãäå
√
D1 = {(x, y) | 0 ≤ y ≤ 1, 0 ≤ x ≤ y}
D2 = {(x, y) | 1 ≤ y ≤ 2, 0 ≤ x ≤ 2 − y}
ZZ
ZZ
D
Z1
=
D1
√
Zy
dy
0
Ïðèìåð 45.
ZZ
f (x, y) dxdy = f (x, y) dxdy + f (x, y) dxdy =
D2
ZZ
1
0
f (x, y) dx + dy
0
Âû÷èñëèòü
Z2
2Z−y
x
y dxdy, åñëè îáëàñòü D îãðàíè÷åíà ëèíèÿìè
( +2 )
D
y = x2 , y2 = x.
y
y = x2
1
O
ZZ
Z1
x
y dxdy = dx
√
Zx
( +2 )
D
0
Z1
=
0
√
x
y dy =
y2 = x
x
1
Z1
( +2 )
√
x
xy + y2 x2 dx =
0
x2
x x + x − x3 − x4 ) dx =
(
f (x, y) dy.
2
1
1
1
1
x5/2 + x2 − x4 − x5 5
2
4
5
0
=
2
5
+
1
2
−
1
4
−
1
5
=
9
20
.
21
Âû÷èñëèòü
Ïðèìåð 46.
Z
dl
p
AB
x2 + y2 + 1
AB îòðåçîê ïðÿìîé, ñîåäèíÿþùåé òî÷êè A(0, 0) è B(1, 1). Ñîñòàâèì
óðàâíåíèå ïðÿìîé, ïðîèñõîäÿùåé ÷åðåç òî÷êó A è òî÷êó B: y = x. Òîãäà
p
√
dl = 1 + (y0 (x))2 dx = 2 dx. Òàêèì îáðàçîì,
ãäå
Z1
Z
AB
√
=
d(
(
Ïðèìåð 47.
√
0
x
2 )
q√
√
dl
2dx
2dx
p
√
=
= √
=
x2 + y2 + 1
x2 + x2 + 1
2x2 + 1
x
2 )2 + 1
=
√ 1
√
√
arctg 2x = arctg 2 − arctg 0 = arctg 2.
0
Âû÷èñëèòü
Z
L
(
x2 − xy) dx + (y2 − xy) dy,
ãäå L ïàðàáîëà y = x2 , 0 ≤ x ≤ 1.
y = x2 , dy = 2xdx
Z
L
2
2
(x − xy) dx + (y − xy) dy =
Z1
0
Z1
0
(
x2 − xx2 ) dx + ((x2 )2 − xx2 )2xdx =
x2 − 2x3 + x4 )2xdx = 2
Z1
(
4
x
4
−2
x5 x6 1
5
+
6
0
=
0
1
4
(
x3 − 2x4 + x5 ) dx =
−
2
5
+
1
6
=
1
60
.
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