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7!
4!
3! 4!5!
n!
(n+ 1)!
(n!)2
n n! .
210
6
5
1
n + 1
(n 1)!
11
8
73
2018
10
7
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A= (3, 1] (0, 2] B = (, 1] [1, +)
A= [0, 1) [2, 3) B= (, 3) [2, 4) A= (, 3] (1, +) B= [1, 3] (2, 4)
A= (, 1) (1, +)
B= [0, 4] (0, 3)
max A= 2 min A max B= min B= 1
max A min A= 0 max B min B= 2
max A= 3 min A max B min B= 1
max A min A max B min B
A =
(n,
n); n= 1, 2, 3, . . .
B =
n, 12n
; n= 0, 1, 2, . . .
C =
n,
1
n 1
; n= 1, 2, 3, . . .
D =
n,
n
n+ 1
; n= 1, 2, 3, . . .
.
sup A = + infA = min A = 1 max A A sup B = 0 infB = min B =1 max B B sup C = max C = 0 infC =1 min C sup D = 1 infD =min D= 1/2 max D
0 5 10 15 200
1
2
3
4
5
A
x
y
0 5 10 15 201
0.5
0
0.5
1
B
x
y
0 5 10 15 201
0.8
0.6
0.4
0.2
0
C
x
y
0 5 10 15 200
0.2
0.4
0.6
0.8
1
D
x
y
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{an} n1
an= (
2)n
10
an= (1)nn0 an= ln
n 1n
0 an= 10
1/n >1
n {an} n
n {an} n
nN an0 n 1n
1 1n
>0
{bn} n1
n
bn= 3n 100
9
bn= 3 ln n < 0
bn= 2n 1000
bn= e
n 100< 0
3n 1009
nlog3100
9 n3
ln n > 3 n > e3 n21 nlog2100 n7
en
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limn
21/n = 1
limn ln
n+ 1
= +
> 0
ln
1 +1
n
< n > N
ln1 + 1n< 1 + 1
n
< e n > (e
1)1 =N
>0
11 n2
< 1n2 1 < n > N
n >
1 + 1 =N
> 0 |21/n 1|< n > N 21/n 10 n 21/n < + 1 n > (log2( + 1))
1 =N
M >0 ln
n + 1
> M
n+ 1 > eM
n >(eM 1)2 =N
an= n2 +n +
M > 0 an > M
n2 + n M >0 n >1 +
1 + 4M
2 =N
{an} {an} {an}
limn
an= l, an> 1 nl >1 anan+1 n {an}
an+1an n lim
nan=
an = (1)n 1 1 an = 1 +
1
n an > 1 n l= 1
an =
1
n
{an} {bn} limn anbn= 0
an= n sin(
2 +n)
an= ln n cos(n)
an= (1)n(n n2)
an= n+ (1)nn
sin(2 +n) = (1)n an = (1)n n + (M, +) M >0 n bn = 1/n2
cos(n) = (1)n an = (1)n ln n bn = 1/ ln
2 n
(1)n+1n2 bn = 1/n
3
+
bn = 1/n2
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an= n+
(1)nn
sup inf max min
sup an = + infan= min an = 0 max an limn an = +
an limn an = 0
limn 2an = 1
limn(an+1 an) = 0 limn 1an = +
supnN an< +
an = 1
2
n
kN nsin
k n
nN ksin
k n
k= 1 k >1 n+ n = mk m N n= 2mk+ 1 sin
k= 0
nN limk sin
k n
= sin 0 = 0
3
2n4 + 3n3 + 1
n+ ln n
n nn+en
ln n nn ln n
n1/2 +n1/3 + 1
n1/4 +n1/5
3
2n4 + 3n3 + 1
n + ln n 2
1/3n4/3
n =
3
2n
n nn + en
nn
= 1
ln n nn ln n
nn
=n
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n1/2 + n1/3 + 1
n1/4 + n1/5 n
1/2
n1/4 = 4
n
limn
1 +n
n n2 3nn
limn
1
1 + (
1)nn
limn
n ln nn
limn
n n
limn
n
1 (2/3)n lim
n
2nen en3n lim
nen 2n
3n
limn
3n sin(n/2)
2n
1 + n
n n2 3nn
n7/3
n =n4/3 lim
n(n4/3) =
limn
|1 + (1)nn|= + limn
1
1 + (1)nn = 0
n ln nn
n limn
n= +
n n n
1 (2/3)n 1 limn
n
1 (2/3)n = +
2< e
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q >1 limn1
qn
nk=0
qk
nk=0
qk = 1 qn+1
1 q ;
limn1
qn
nk=0
q
k
= limn1
qn+1
qn(1 q) = limnqn(qn
q)
qn(1 q) = limn q
1 q = q
q 1
q >1
1
+ 1
an= 1 +(1)n
n
an= (1)nn an= n + (1)n
an=1 1n
{an} lim
nan3n
= 0, limn
ann3
= +
limn
ann
= +, limn
ann
n= 0
limn a
n2n
= +, limn a
n3n
= 0
limn
anln n
= 0, limn
a2nln n
= +
an= n4
an= n 3
n
an= en
an= (ln n)2/3
a, bR+
a, b
an=2a+b 1a
n
ab (a, b)
2a + b 1a
1 0a + b 10.
A(1/3, 0) B(1, 0) C(0, 1) AB AC BC
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n=0
n
n!
n=1
(1 en)
n=1
21/n
nen
n=1
(1)n nn
n=0
1 n+n21 +n2 +n4
n=1
(1)n 1n2 +n
n=2
(1)nn ln n
n=1
3 +en
n!
n=0
en +
1
2
n
n=0
an limn
an= 0
n + 1
(n + 1)! n!
n=
1n(n + 1)
n0
limn(1en) = 1 +
n2
2n
ne
n 1e
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n=1
1
n2
1n2 + n
1
n ln n
3
n=0
1
n!
n=0
1
2
n
n=21
ln n
n=1
1
n2 + sin n
n=1
(1)n 1n1/3
n=1
ln n nn+ 1
n=21
ln n >
n=21
n
n=1
1
n2
1
n1/3
ln n nn + 1
1n
n=1
1
n1/2
n=0
n2 + 1
n!
n=1
1
n+ ln n
n=0
(1)n(1 +en)
n=0
n+ sin n
1 +n2
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(n + 1)2 + 1
(n + 1)! n!
n2 + 1 =
n2 + 2n + 2
n3 + n2 + n + 1 1
nn0
n=11
n
limn(1 +
en) = 1 n=0
(1)n
n=1
1
n
n=1
1
nlnn
n=2
1(ln n)n
n=1
1
nlnn 1 qn
1 + qn qn
qn = 1
n=0
1 = + 0< q
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n=1
1
n 1
n + 3
=
1 14
+
1
2 1
5
+
1
3 1
6
+
1
4 1
7
+
1
5 1
8
+ ...=
= 1 +1
2+
1
3+
1
4 1
4
+
1
5 1
5
+ ...= 1 +
1
2+
1
3=
11
6 .
a
b
a
b
n=1
na
1 +nb
n=1
1
nba
ba > 1 b > a+ 1 + b < a+ 1 a, b b > a + 1
an= sin(n/2) + | cos(n/2)|
limn an
n=1an
1;1;1; 1;1; ... limn an
n=1
1
n1+ n1 0< 1
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1
1
ln x
1x x3
11 x2
x2 4x+ 3
ln
1
1 |x|
11 ln |x|
(0, e) (e, +) (, 1) (1, 0) (0, 1) (1, +) (1, 1)
(, 1) [3, +) (1, 1)
(e, 0) (0, e)
arcsin
1
x2
arccos(2
x2)
(, 1] [1, +) [3, 1] [1, 3]
1 xex
ex ex
sin x
1 +x2
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x x2
x2 x3
ex +ex
x|x| sin x cos x
1 x x2 (x 2)3
1 +x+ cos x
|x3| ln x ex
1 x2
ex x sin(x /4)
ln x sin x 1 e|x|
1 (x+ 1)2 ln x x
1/
x 1/x 1/x2 (0, 1]
ex/2
ex
e2x
sin(x/2) sin x sin(2x) [0, 4] ln(2x) ln x ln(x/2) (0, +)
f1(x) = x2 1
f2(x) = (x+ 1)2 1
f3(x) =|x2 1|
f4(x) = 1 (x 1)2
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4 2 0 2 420
15
10
5
0
51xx
2
x
y
2 0 2 4 6100
50
0
50
100(x2)
3
x
y
10 5 0 5 1010
5
0
5
10
151+x+cos(x)
x
y
4 2 0 2 40
10
20
30
40
50
60
70|x
3|
x
y
0 0.5 1 1.5 27
6
5
4
3
2log(x)e
x
x
y
4 2 0 2 415
10
5
0
51x
2
x
y
4 2 0 2 40
10
20
30
40
50
60e
xx
x
y
5 0 51
0.5
0
0.5
1
sin(x/4)
x
y
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0 2 4 6 82
1
0
1
2
3log(x)sin(x)
x
y
4 2 0 2 40
0.2
0.4
0.6
0.8
11e
|x|
x
y
10 5 0 5 10120
100
80
60
40
20
0
201(x+1)
2
x
y
0 2 4 6 87
6
5
4
3
2
1log(x)x
x
y
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
1/x1/2
, 1/x, 1/x2
x
y
1/x1/2
1/x
1/x2
1 0.5 0 0.5 10
2
4
6
8
x
y
ex/2
, ex, e
2x
ex/2
ex
e2x
5 0 52
1
0
1
2
3
4sin(x/2), sin(x), sin(2x)
x
y
sin(x/2)sinxsin(2x)
0 0.2 0.4 0.6 0.8 18
6
4
2
0
2
x
y
log(x/2), log(x), log(2x)
log(x/2)log(x)log(2x)
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2 1 0 1 21
0
1
2
3x
21
x
y
2 1 0 1 22
0
2
4
6
8(x+1)
21
x
y
2 1 0 1 20
0.5
1
1.5
2
2.5
3|x
21|
x
y
2 1 0 1 28
6
4
2
0
21(x1)
2
x
y
f f
f(x) =
x sex[0, 1]2x 1 sex(1, 2]
f(x) =
x se x[1, 0]2x se x(0, 1]
f(x) = x se x[1, 0]x/2 se x(0, 1]
f(x) =
x+ 1 se x[1, 0]1 +x/2 se x(0, 1] .
f1(y) =
y se y[0, 1](y+ 1)/2 se y(1, 3]
f1(y) =
y se y[0, 1]y/2 se y[2, 0)
f1(y) =
y se y[1, 0]2y se y(0, 1/2]
f1(y) =
y 1 se y[0, 1]2y 2 se y(1, 3/2]
f1(x) =
x/2 x02x x < 0;
f2(x) =
x/2 + 1 x02x 1 x < 0.
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1 0 1 2 3 41
0
1
2
3
4(a)
x
y
3 2 1 0 1 2 33
2
1
0
1
2
3(b)
x
y
2 1 0 1 22
1
0
1
2(c)
x
y
2 1 0 1 22
1
0
1
2(d)
x
y
f11 (y) =
2y se y0y/2 se y
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f(x) = e2x3
f(x) =
1 2x
f(x) = arctg(2x 1) f(x) = ln(2 + 3x)
f : R
(0, +
) f1 : (0, +
)
R f1(y) = ln y+ 3
2
f : (, 1/2][0, +) f1 : [0, +)(, 1] f1(y) = (1 y2)/2 f : R (/2, /2) f1 : (/2, /2)R f1(y) =tan y+ 1
2
f : (2/3, +)R f1 : R (2/3, +) f1(y) = ey 2
3
1 0 1 2 3 41
0
1
2
3
4(a)
x
y
2 1 0 1 2 32
1
0
1
2
3(b)
x
y
2 1 0 1 2
2
1
0
1
2(c)
x
y
2 1 0 1 2 3
2
1
0
1
2
3(d)
x
y
limx+
x sin x
1 +x2
limx+
x 3x1 +x
limx0
x2
1 ex
limx1
ln x
ex e
x+
x sin x
1 + x2 sin x
x
limx+
sin x
x = 0
x+
x 3x1 + x
1x
limx+
1x
= 0
x0 1 ex x x2
1 ex x
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x 1 =y
limx1
ln x
e(ex1 1)= limy0ln(y+ 1)
e(ey 1) .
y0 ln(y + 1)y ey 1y ln(y+ 1)e(ey 1)
1
e
1
e
limx0+x ln(x)
limx1
(x 1)2sin(1 x)
limx0
tg x
x2
limx
x4ex
limx1
1 x2 ln x
limx+
arccos
1
x
arctg x
limx0
e3x 1ln(1 2x)
limx
2
x 2cos x
x= y lim
x0+
x ln(
x) = lim
y0+y ln y = 0
y= 1 x limx1
(x 1)2sin(1 x) = limy0
y2
sin y= lim
y0y = 0
x
0 tan x
x lim
x0
tg x
x2
= limx0
1
x
=
x= y lim
xx4ex = lim
y+y4
ey = 0
x 1 =y
limx1
1 x2 ln x
= limy0
y2ln(1 + y)
= limy0
y2y
=12
;
limx+
arccos
1
x
arctg x
=arccos 0
/2 =
/2
/2= 1
x0 e3x 13x ln(1 2x) 2x 32
x
/2 =y
limx
2
x 2
cos x = lim
y0y
cos(y+ /2)= lim
y0y
sin y =1.
x+
1
2 + sin x
ln x sin x
x2 sin x
cos2 x
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xn = (2n + 1)/2
xn = n/2
xn = (2n + 1)/2
xn = n/2
limx0sin x
ln(1 +x2)
limx
2
x 2cos2 x
limx0
xe1/x
limx0
arctg(1/x)
x2
x0 sin xx ln(1 + x2)x2 limx0
sin x
ln(1 + x2) = lim
x01
x =
x
2
=y
limx
2
x 2
cos2 x = lim
y0y
cos2(y+ /2)= lim
y0y
sin2 y = lim
y01
y =;
1
x=y lim
yey
y lim
y+ey
y = + lim
yey
y = 0
x0 arctg(1/x) /2 1x2
+ limx0
arctg(1/x) 1x2
=
limx0
sin(2x)
1 e3x
limx0sin(2x)
|1 e3x|
limx0
ex e2xx
limx0
x
ln(1 +x2)
limx0
sin x2 sin xx2 x
limx0
| sin x|x
limx0+
xe1/x
sin(2x)
1 e3x 2
3
2/3
|1 e3x|=
1 e3x x0e3x 1 x < 0
2/3 2/3
x0 ex(1 ex)
x ex 1
x
ln(1 + x2) 1
x
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sin x2 sin xx2 x = =
sin x2
x2 x
x 1sin x
x 1
x 1 1 1
y= 1/x
limx0+
xe1/x = limy+
ey
y = +.
x0+
x
x+ tg 3
x
x0 x + tg 3xx + 3x 3x x/ 3x= 6x
limx+
1 +
1
2x2
3x2
limx+
1 +
1
2x2
3x2= lim
x+
1 +
1
2x2
2x23/2 2x2 = y
limy+
1 +
1
y
y3/2=e3/2 =e
e.
3x2 + 1
1 x
x2 +ex
2x+ 3
x2 1ln |x| 2x
x x1/31 +ex
y =3x 3
+ y = 1/2x 3/4
y = 0 x+ x
f(x) = x(1 +ex) + ln x x+
limx+
f(x)
x = 1 lim
x+(f(x) x) = lim
x+xex + lim
x+ln x = +
n=1
1 cos
1
n
1 cos xx2
2 x0
1 cos1n 1
2
1
n
2=
1
2n2 pern ;
1
2
n=1
1
n2
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ln3(1 +x2)
x
tg x
sin2(3 2x)
ex
ln x
cos3(1 x2)
ln x
1 x2 exx
2
x
1 + sin x
ln3(1 + x2)
=
6x ln2(1 + x2)
1 + x2
x
tg x
=
tg x x(1 + tg2 x)tg2 x
sin2(3 2x)
=4 sin(3 2x)cos(3 2x)
ex
ln x
=ex x ln x 1
x ln2 x
cos3(1 x2)
= 6x cos2(1 x2) sin(1 x2)
ln x
1 x2
=1 x2 + 2x2 ln x
x(1 x2)2
ex
x2
=ex
x2
(1 2x)
x
1 + sin x
=
1 + sin x x cos x(1 + sin x)2
D
x tg(1 +x2)
D
arcsin(1 + 3x3)
D
1 + sin x
1 cos2 x
D
ln
ln(x+ 1)
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D
x tg(1 + x2)
= tg(1 + x2) + 2x2(1 + tg2(1 + x2))
D
arcsin(1 + 3x3)
= 9x21 (1 + 3x3)2
D
1 + sin x
1 cos2 x
=cos x(2 + sin x)sin3 x
D
ln
ln(x + 1)
= 1
(x + 1)ln(x + 1)
D
2x2
D x
ln2 x
D
x2 sin x
D sin(ex)
D2x2= 2x2 ln 2 2x D
x
ln2 x=
ln x 2ln3 x
D
x2 sinx
= x2 sinx
2cos x ln x + 2sin x
x
D sin
ex
=ex cos ex
f
f(x) = ex
2
x=
1
f(x) =
1
1 x x= 0
f(x) =
1
tg x x= /4
f(x) = 1
ln x x= e
f(x) x0 y f(x0) = f(x0)(x x0)
y= 2x/e + 3/e
y= x + 1
y=2x + /2 + 1
y= 2 x/e
(0, +) f(x) = ln x g(x) = arctg x
x0 f g
x0, f(x0)
x0, g(x0)
f (x0, f(x0)) f(x0) = 1/x0
g (x0, f(x0)) g(x0) = 1/(1 + x20)
1
x0=
1
1 + x20 x20x0+ 1 = 0
x0
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f(x) = x3 x0 [0, 1]
f
(x0, f(x0)) 1
(x, f(x)) f(x) = 3x2 f(x) = 1
x = 13
x0 = 1
3
1
3,
1
3
3
y =
x 23
3
D 1
f
g(x)
D
f(x)g(x)
D
1
f(x)g2(x)
D
f
1
g(x)
D 1
f
g(x) =f(g(x)) g(x)
f2(g(x))
D
f(x)g(x)
= f(x)g(x)
g(x) ln f(x) + g(x)f(x)f(x)
D 1
f(x)g2(x)=f
(x)g(x) + 2f(x)g(x)f2(x)g3(x)
D
f
1
g(x)
=f
1
g(x)
g
(x)g2(x)
D
f(1 +x+g(x))
D
f(1 +x) g(1 x)
D
f
g(x) g(x+ 1) D
f(xg(x))
D
f(1 + x + g(x))
= f(1 + x + g(x)) (1 + g(x))
Df(1 + x) g(1 x)= f(1 + x) g(1 x) f(1 + x)g(1 x)
D
f
g(x) g(x + 1)
= f (g(x) g(x + 1)) (g(x) g(x + 1))
D
f(xg(x))
= f(xg(x)) (g(x) + xg(x))
D
f(x)g(x)h(x)
D
f(x)g(x)h(x)
= f(x)g(x)h(x)
g(x)h(x) + g(x)h(x)
ln f(x) + g(x)h(x)f(x)f(x)
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f(x) = xex
f
f
0 f1(y) y= f(0)
(, 0] [0, 1] [1, +) f1(0) =
0
f1
(0) = 1
f(0) = 1
2 1 0 1 2 3 410
9
8
7
6
5
4
3
2
1
0
1
xex
x
y
f(x) = x3 +x f1(y)
f1(2)
f1
(2)
f(x) = 3x2 + 1 > 0x R f(x) R f1(y) R
f1
(y) = 1
f(x) x =
f1
(y)
f(x)
f1
(2)
x
3
+ x 2 = 0
x= 1 f1(2) = 1
f1
(2) = 1
f(1) =
1
4
sin |x|
3
x 1 x|x 1|
| sin x|
x= 0 D (sin |x|) = cos |x| D(|x|) |x| x= 1 3
y y = 0
x= 1 |y| x= k kZ y= sin x
f(x) =
1 x
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f R x=1 x=1 g x= 1 x=1 x=1
3 2 1 0 1 2 3
4
2
0
2
4
6
f
x
y
3 2 1 0 1 2 3
4
2
0
2
4
6
g
x
y
x= 1
x < 0 x > 0
(0, 1)
R
f(x) =|x 1| f(x) = arctg x
f(x) = 1/x
f(x) = 11 + x2
(, +)\{0} 0 1 f(x) =|x 1| |x + 1| + sgn x
2 1.5 1 0.5 0 0.5 1 1.5 21
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4|x1||x+1|+sign(x)
x
y
f
f >0 f(0)< 0 f(1)> 0
f(x) = x2 x + 1
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f
f(x) =ex
f
g
f(x) =
aebx sex01 +x2 sex
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x= 0 x= 0 x+ y = 0 x x =2 e2/4
R
x+ y = 0 x x= 3/2
1
2e3 x= 2
R x y = 0 x+ x=2 4/e2 x= 0
x=2 2 R x y = 0 x+
x= 1/2 1/2e x= 0
4 2 0 2 4
0
2
4
6
8
101/(x
2e
x)
x
y
2 1 0 1 2
10
8
6
4
2
0
(x1)/e2x
x
y
1 0 1 20
2
4
6
8
10x
2e
x
x
y
1 0 1 22
0
2
4
6
8
10(x1)e
2x
x
y
e2x ex
x2 ln x
ln(x+ 1) x arctg(1 x)
R y = 0 x x = ln 2 x= ln 4 x > ln 4 x 1 x= 1
1
x 1
x 1
x
1 + |x|
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1 0 1 210
0
10
20
30
40
50e
2xe
x
x
y
0 0.5 1 1.5 20
1
2
3
4
5x
2log(x)
x
y
1 0 1 2 3 44
3
2
1
0
1log(x+1)x
x
y
4 2 0 2 41.5
1
0.5
0
0.5
1
1.5arctg(1x)
x
y
ln(1 +x2)
1
x+ ln x
1
x 1
x 1= 1
x(1 x) xR \ {0, 1} x= 0x= 1 y = 0 x= 1/2 (0, 1)
R
y = 1 + y =1
x >0
x < 0
R x = 0 x=1 (1, 1)
(0, +) x= 0 x= 1 x= 2 x 2
1 x 2x2
exx
x
ln x
x(2 x)ex
[1, 1/2] (1/2, 1/2) x=1/4 x=1, 1/2
x 0 x > 0 x= 1/4 x= 0
(0, +) \ {1} x = 1 limx0+f(x) = 0 x= e (1, e
2) x= e2
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2 1 0 1 210
5
0
5
101/x1/(x1)
x
y
2 1 0 1 21
0.5
0
0.5
1x/(1+|x|)
x
y
10 5 0 5 100
1
2
3
4
5log(1+x
2)
x
y
0 2 4 6 8 100
2
4
6
8
101/x+log(x)
x
y
R y = 0 + x= 2 2 x= 2 +
2 (3 3, 3 + 3)
x= 3 3
1.5 1 0.5 0 0.5 10.5
0
0.5
1
1.5(1x2x
2)
1/2
x
y
0 1 2 30
0.5
1
1.5
2
2.5
3e
xx1/2
x
y
0 2 4 6 85
0
5x/log(x)
x
y
2 0 2 4 6 83
2
1
0
1
2x(2x)e
x
x
y
1
x 1 x2
1
x2 2x+ 2
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R
y= 0 x= 1/2 (0, 1) x= 0, 1
R
y = 0
x= 1
(0, 2) x= 0, 2
2 1 0 1 22
1.5
1
0.5
0
0.5
11/(x1x
2)
x
y
2 0 2 40
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
21/(x
22x+2)
x
y
f f
f(x) = |x|x 1
f(x) =
|x| 1x
R
\{1}
x= 1 y = 1 +
y=1 x= 0 (0, 1) f x= 0 x= 0
x= 0 x= 0 y = 1 + y =1 x < 0 f
x3 2x2 +x+ 1 1 x+x2 x3
x(4 x+x2
/2) 1 3x+x2 x3
x 2/3 limx
f(x) =
x > 1/3 x 2/3 limx
f(x) =
x > 1/3 x
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4 2 0 2 43
2
1
0
1
2
3|x|/(x1)
x
y
2 1 0 1 22
1.5
1
0.5
0
0.5
1
1.5
2(|x|1)/x
x
y
1 0 1 23
2
1
0
1
2
3x
32x
2+x+1
x
y
2 1 0 1 25
0
5
10
151x+x
2x
3
x
y
1 0 1 2 310
5
0
5
10
15
20x(4x+x
2/2)
x
y
4 2 0 2 4100
50
0
50
10013x+x
2x
3
x
y
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f(x) = ln x ln(x 1) f
(1, +) x = 1 y = 0
f(x) = 1
1 +ex
R y = 0 x
+
y = 1
x x= 0
4 3 2 1 0 1 2 3 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
11/(1+e
x)
x
y
f(x) = x2
a2 x2 a f
f
a
f
a
[a, a] [0, +) x= 0 x=a x=
2/3a a
f(b) f(a)b a = f
(c) c (a, b) c
f(x) = 1 x3 a= 0 b= 1 f(x) = x2 x a= 1 b= 2
f(x) = 1/x3 a= 1 b= 2
f(x) = 1
1 +x
a= 0
b= 2
f(c) =1 3x2 =1 x=1/3 c= 1/3(0, 1) f(c) = 2 2x 1 = 2 c= 3/2 f(c) =7/8 3/x4 =7/8 x= 4
24/7 c= 4
24/7(1, 2)
f(c) =1/3 (1 + x)2 =1/3 c= 3 1
R
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4 2 0 2 40
5
10
15
20
25
a=1
x
y
4 2 0 2 40
5
10
15
20
25
a=2
x
y
4 2 0 2 40
5
10
15
20
25
a=3
x
y
4 2 0 2 40
5
10
15
20
25
a=4
x
y
ex
ex
ex
ex
1 0 1 2 33
2.5
2
1.5
1
0.5
0
ex
x
y
1 0 1 2 30
0.5
1
1.5
2
2.5
3
ex
x
y
3 2 1 0 13
2.5
2
1.5
1
0.5
0e
x
x
y
3 2 1 0 10
0.5
1
1.5
2
2.5
3e
x
x
y
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f
1 1 f(x) =|1 x2|
f
(1, f(1))
f
1 1
f(x) = ex
23x+2
f(x) = ln(2 +x2)
f(x) = sin(x )
f(x) = e3x2
y =x+ 2 f(1) = 3 f 1
y = 2
3(x1) + ln3 f(1) = 2/9 f 1
y = (x1) f(1) = f 1
y=2e2x + 3e2 f(1) = 2e2 f 1
0 0.5 1 1.5 20
0.5
1
1.5
2(a)
x
y
0 0.5 1 1.5 20.4
0.6
0.8
1
1.2
1.4
1.6
(b)
x
y
0.5 1 1.51
0.5
0
0.5
1(c)
x
y
0.5 1 1.50
5
10
15
20(d)
x
y
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sin2(x 1) cos(x 1) dx
ex
(ex 6)4 dx
ln(1 +x)
2(1 +x) dx
arctg2 x
1 +x2 dx
(sin(x 1)) = cos(x 1) sin2(x 1) cos(x 1) dx= sin
3(x 1)3
+ c, cR;
(ex 6) = ex (ex 6)4ex dx= 1
3(ex 6)3 + c, cR;
(ln(1 + x))= 1
1 + x
ln(1 + x)
2(1 + x) dx=
(ln(1 + x))2
4 + c, cR;
(arctg x) = 1
1 + x2
arctg2 x1 + x2 dx=
arctg3 x
3 + c, cR.
1
x ln xdx
2x
1 +x4dx
cos(1/x)
x2 dx
11 3x2 dx
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(lnln x)= 1
ln x 1
x lnln x + c cR
(arctg x2) = 2x
1 + x4 arctg x2 + c cR
sin
1
x
= cos(1/x) 1
x2
sin1x
+ c cR
(arcsin
3x) = 1
1 3x2
3 1
3arcsin
3x + c c
R
x
x2 + 4x+ 3
ln x
x
x sin x
(arcsin x)2
x3
1 +x2dx
1
cos x
x
x2 + 4x + 3=
A
x + 1+
B
x + 3 A=1/2 B= 3/2
12
ln |x + 1| +32
ln |x + 3| + c= ln
(x + 3)3
x + 1 + c, cR;
ln x
x dx= ln2 x
ln x
x dx,
ln x
x dx=
ln2 x
2 + c, cR;
x sin xdx=x cos x +
cos xdx=x cos x + sin x + c, cR;
x= sin y (arcsin x)2dx=
y2 cos ydy,
y2 cos ydy = y2 sin y+ 2y cos y 2sin y+ c, cR
(arcsin x)2dx= x(arcsin x)2 + 2 arcsin x
1 x2 2x + c;
x3
1 + x2
= x(x2 + 1) x
1 + x2
x3
1 + x2dx=
x dx 1
2
2x
1 + x2dx=
x2
2 ln
1 + x2 + c, cR;
t= tgx
2
2
1 t2 dt 2
1 t2 = 1
1 t + 1
1 + t
2
1 t2 dt= ln1 + t
1 t+ c
1
cos xdx= ln
1 + tgx
2
1 tgx2
+ c, cR.
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3x2 ln xdx
1
x+ 1dx
1
1 +exdx
ex sin xdx
3x2 ln xdx= x3 ln x
x2dx= x3 ln x x
3
3 + c, cR;
x= t2 dx= 2t dt
2
t
t + 1dt= 2
dt 2
1
t + 1dt= 2t ln2(t + 1) + c, cR,
1
x + 1dx= 2
x ln2(x + 1) + c;
ex =t dx= 1/t dt 1
t(1 + t)dt=
1
t 1
t + 1
dt= ln
tt + 1
+ c, cR,
1
1 + exdx= ln
ex
ex + 1+ c;
2
e
x
sin xdx= e
x
(sin x cos x)
ex sin xdx=ex(sin x cos x)
2 + c, cR.
sin2 x cos2 x dx
x2
1 +xdx
sin x cos x= (sin 2x)/2 2x= y sin2 x cos2 x dx=
1
4
sin2 2x dx=
1
8
sin2 y dy,
sin2 x cos2 x dx=
1
8
y2
4 sin y cos y
2
+ c=
1
8
x2 1
2sin 2x cos2x
+ c, cR;
x2 = (x + 1)(x 1) + 1
x2
1 + xdx =
(x 1) dx +
1
1 + xdx =
x2
2 x + log |x + 1| + c, cR.
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(x 2)2 ln x dx
x2 sin x dx
x arctg x dx
ln(1 +x2) dx
(x 2)2 ln x dx =(x 2)
3
3 ln x
(x 2)3
3 1
xdx
=(x 2)3
3 ln x 1
3
x2 6x2 + 12x 8
x
dx
=(x 2)3
3
ln x
1
3x
3
3 2x3 + 6x2
8 ln x+ c
=x3 6x2 + 12x
3 ln x +
5
9x3 2x2 + c, cR;
x2 sin x dx =x2 cos x +
2x cos x dx
=x2 cos x + 2x sin x 2
sin x dx
=x2 cos x + 2x sin x + 2 cos x + c, cR;
x arctg x dx =
x2
2 arctg x
x2
2(1 + x2)dx
=x2
2 arctg x 1
2
1 1
1 + x2
dx
=x2
2 arctg x 1
2(x arctg x) + c, cR;
ln(1 + x2) dx =x ln(1 + x2) 2 x2
1 + x2dx
=x ln(1 + x2) 2
1 11 + x2
dx
=x ln(1 + x2) 2 (x arctg x) + c, cR.
sin
1x
x2
dx
1
3
x(1 + 3
x2)dx
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1
2 x2 dx
x22xdx
x1 =t
sin
1x
x2 dx=
sin t dt= cos t + c= cos1
x+ c, cR;
x= t3 1
3
x(1 + 3
x2)dx=
3t
1 + t2dt=
3
2
2t
1 + t2dt=
3
2ln(1+t2)+c=
3
2ln(1+
3
x2)+c, cR;
1
2 x2 = 1
(
2 x)(2 + x) = 1
2
2
12 x +
12 + x
,
12 x
2dx=
1
22ln
2 + x
2 x + c, c
R;
x22xdx =
2x
ln 2x2
2x
ln 22x dx
= 2x
ln 2x2 2
ln 2
2x
ln 2x
2x
ln 2dx
= 2x
ln 2x2 2
x+1
ln2 2x +
2x+1
ln3 2+ c, cR.
2 x2dx
1x(1 +x)
dx
sin
x
dx
1
2 cos x dx
x=
2sin t dx=
2cos t dt
2 x2 = 2cos t
2 x
2
dx= 2
cos2
t dt= t + sin t cos t + c= arcsin
x
2 +x
2
2 x2
+ c, cR;
x= t2 1x(1 + x)
dx= 2
1
1 + t2dt= 2 arctg t + c= 2 arctg
x + c, cR;
x= t2 sin
x dx = 2
t sin t dt= 2
t cos t +
cos t dt
= 2t cos t + 2 sin t + c=2x cos x + 2 sin x + c, cR;
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t= tanx
2 x= 2 arctant dx=
2
1 + t2dt cos x=
1 t21 + t2
1
2 cos x dx= 2
1
3t2 + 1dt= 2 arctan
3t + c= 2 arctan
3tan
x
2
+ c, cR.
ex ex
ex +exdx
ex ex
ex + exdx=
sinh x
cosh xdx = ln(cosh x) + c, cR.
21
1x(1 +
x)
dx
10
x1 +x2
dx
1
0
6x+ 3x2 + 1 +x
dx
10
x3
1 +x4dx
x= t2 dx= 2t dt
2
21
1
1 + tdt=
ln(1 + t)2
21
= ln3 + 2
2
4 ;
1
2 10
2x
1 + x2 dx= 1 + x21
0 =
2 1;
3
10
2x + 1
x2 + 1 + xdx=
ln(x2 + x + 1)3
10
= ln27;
1
4
10
4x3
1 + x4dx=
ln
4
1 + x410
= ln 4
2.
10
1
1 +
4xdx
10
x arctg xdx
/20
sin2 x x sin x2
dx
1/20
x1 x2 dx
4x= t2 10
1
1 +
4xdx=
1
2
20
t
1 + tdt=
1
2
t ln |t + 1|
20
= 1 ln
3;
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x2
2 arctg x
10 1
2
10
x2
1 + x2dx=
8 1
2
10
1 1
1 + x2
dx=
8 1
2
x arctg x
10
=
4 1
2;
/2
0
sin2 xdx 12
/2
0
2x sin x2dx= x sin x cos x
2 /2
0+
1
2cos x2/2
0=
4 1
2;
12
1/20
2x1 x2 dx=
1
2
2
1 x21/20
= 1
3
2 .
10
ex
1 +exdx
/20
sin x
1 + sin xdx
2
1
1
x2
+x
dx
10
x
ex2dx
ex =t e1
1
1 + tdt=
ln |t + 1|
e1
= lne + 1
2 ;
t= tgx
2
1
0
2t
1 + t2 1
1 + 2t
1 + t2
21 + t2
dt=
1
0
4t
(1 + t2)(t + 1)2dt;
4t
(1 + t2)(t + 1)2 =
A
t2 + 1+
B
(t + 1)2
A= 2 B=2
2
10
1
t2 + 1dt 2
10
1
(t + 1)2dt= 2
arctg t
10
+ 2
1
t + 1
10
=
2 1;
1
x2 + x =
1
x 1
x + 1
21
1
x2 + x dx=
ln x21 ln(x + 1)
2
1 = ln
4
3 ;
x
ex2 =xex
2
=12
ex
2
12
10
ex
2
dx= 1 e1
2 ;
x2 =t
21
x3 +x
dx
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10
3x
4 x2 dx
21
x
x+ 1dx
1/30
x
1 4x2 dx
3 + x= t 21
x3 + x
dx=
54
t 3t
dt=
54
(t1/2 3t1/2) dt=
2
3t
t 6
t54
=4
3(5 2
5);
x= 2sin t dx= 2cos t dt
4 x2 = 2 cos t 10
3x
4 x2 dx= 24 /60
cos2 t sin t dt=24
cos3 t
3
/60
= 8 3
3;
x= t2
2
1
x
x + 1dx =
2
1
2t2
t2 + 1dt = 2
2
1 1 1
t2 + 1 dt= 2 [t arctg t]21 = 2(2 arctg 2 1 + /4);
(1 4x2) =8x 1/30
x
1 4x2 dx = 18
1/30
8x
1 4x2 dx
= 18
2
3(1 4x2)3/2
1/30
= 1
12
1 5
5
27
.
10
2x2 31 +x2
dx
/40
sin x+ cos xcos3 x
dx
94
x
1 x dx
2x2 31 + x2
= 2 51 + x2
10
2x2 31 + x2
dx=
10
2 5
1 + x2
dx= [2x 5 arctanx]10 = 2
5
4;
/40
sin x + cos xcos3 x dx = /40
cos3 x sin x dx + /40
cos2 x dx
=
1
2cos2 x + tan x
/40
=3
2;
x= t2 94
x
1 xdx = 2 32
t2
t2 1dt =2 32
1 +
1
t2 1
dt
= 2 32
1 +
1
2
1
t 11
2
1
t + 1
dt =
32
2 1
t 1+ 1
t + 1
dt
=
2t ln |t 1| + ln |t + 1|
3
2=2 + ln2
3.
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f g
f(x) = sin x g(x) = 2 sin x x[0, /2] f(x) =
1
1 +x2 g(x) = 2 +x x[0, 1]
f(x) = ln x g(x) = x+ 1 x[1, e] f(x) = 2x g(x) = 3x x[0, 1]
A=
/20
2(1 sin x)dx= 2
x + cos x/20
= 2
A=
10
2 + x 1
1 + x2
dx=
2x +
x2
2 arctg x
10
=5
2
4
A =
e1
(1 + x ln x)dx =
x+x2
2
e1
e1
ln xdx
ln xdx= x ln x x A= 2x +x2
2 x ln x
e
1
=e2
2
+ e
5
2
A=
10
2x 3x
dx=
2x
ln 2 3
x
ln 1/3
10
= 1
ln 2 2
ln27
0 0.5 1 1.5
0
0.5
1
1.5
2a
x
y
fg
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4b
x
y
fg
1 1.5 2 2.50
1
2
3
4c
x
y
fg
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2d
x
y
fg
D={(x, y); x0, x2 y2x2, y2}
D
x
D={(x, y); 0y2,
y/2xy}
A(D) =
20
yy/2
dxdy =
20
y
y/2
dy=
1
2
2
20
ydy
=
1
2
2
2
3
y
y20
=4
3(
2 1).
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+1
1
(x+ 1)2dx
+
1
1
x(x+ 1)dx
+1/5
e5xdx
0
e1xdx
+1
1
(x + 1)2dx = lim
r+
r1
1
(x + 1)2dx = lim
r
1
x + 1
r1
= 1
2
1
x + 1
+1
limr
1
x + 1
r1
+
1
1
xdx
+
1
1
x + 1dx= ln x
x + 1+
1
= ln2
t=5x 15
1
etdt=1
5
et1
= 1
5e
1 x= t +1
etdt= +
21
1x 1 dx
30
1
x 3 dx
2
0
1(x 2)2 dx
10
e1/x
x2 dx
21
13
x 1 dx
21
(x 1)1/2dx= 2
x 121
= 2
30
1
x 3 dx= ln |x 3|3
0 =
20
1
(x 2)2 dx= 1
x 220
= +
1
x =t
+1
etdt=
et+1
= +
21
13
x 1 dx= 3
2
(x 1)2/3
21
= 3
2
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d
dx
x0
et2
dt
F(x) =
x0
et2
dt
f(x) = ex2
F(x) = f(x) = ex2
f 11f(x) dx= 0
f
10
f(x) dx=
21
f(x) dx=
32
f(x) dx= . . .= 0 ;
f 11f(x) dx= 0
f(x) =
1 |x| 1/21 1/2
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f(x) = x
1 +x2 [0, 1]
f [a, b]
M(f) = 1
b a
ba
f(x)dx,
10
x
1 + x2dx=
1
2
2x
1 + x2dx=
1
2ln(1 + x2)1
0= ln
2;
M(f) = ln
2