Тодорхой интеграл

1. Òîäîðõîé èíòåãðàë
Èíòåãðàë òîîëîë
Òîäîðõîé èíòåãðàë
Ä. Áàòò°ð1
1Department of Computer Science
Ulaanbaatar University
2014 îíû 3-ð ñàðûí 10
ÌÀÒÅÌÀÒÈÊ-2

2. Òîäîðõîé èíòåãðàë
Àãóóëãà
1 Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
ÌÀÒÅÌÀÒÈÊ-2

3. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Àãóóëãà
1 Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
ÌÀÒÅÌÀÒÈÊ-2

4. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
f(x) ôóíêö [a, b], (a < b), õýð÷èì äýýð òàñðàëòã³é
á°ã°°ä F(x)-ò³³íèé ÿìàð íýãýí ýõ ôóíêö, °°ð°°ð
õýëáýë
∃F (x) = f(x), ∀x ∈ [a, b] ⇐⇒ f(x)dx = F(x) + C
áàéã.
Òîäîðõîéëîëò
[a, b] õýð÷ìýýð f(x) ôóíêöýýñ àâñàí Íüþòîí-Ëåéáíèöèéí òîäîðõîé
èíòåãðàë ãýæ F(x) ôóíêöèéí [a, b] õýð÷èì äýýðõ óòãûí °°ð÷ë°ëò
[F(b)-F(a)] òîîã íýðëýõ áà
b
a
f(x)dx = F(b) − F(a), ýíä F (x) = f(x) (1)
ãýæ òýìäýãëýíý.
x = a öýã íü èíòåãðàëûí äîîä õèë, x = b öýã íü èíòåãðàëûí
äýýä õèë
ÌÀÒÅÌÀÒÈÊ-2

5. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
f(x) ôóíêö [a, b], (a < b), õýð÷èì äýýð òàñðàëòã³é
á°ã°°ä F(x)-ò³³íèé ÿìàð íýãýí ýõ ôóíêö, °°ð°°ð
õýëáýë
∃F (x) = f(x), ∀x ∈ [a, b] ⇐⇒ f(x)dx = F(x) + C
áàéã.
Òîäîðõîéëîëò
[a, b] õýð÷ìýýð f(x) ôóíêöýýñ àâñàí Íüþòîí-Ëåéáíèöèéí òîäîðõîé
èíòåãðàë ãýæ F(x) ôóíêöèéí [a, b] õýð÷èì äýýðõ óòãûí °°ð÷ë°ëò
[F(b)-F(a)] òîîã íýðëýõ áà
b
a
f(x)dx = F(b) − F(a), ýíä F (x) = f(x) (1)
ãýæ òýìäýãëýíý.
x = a öýã íü èíòåãðàëûí äîîä õèë, x = b öýã íü èíòåãðàëûí
äýýä õèë
ÌÀÒÅÌÀÒÈÊ-2

6. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
f(x) ôóíêö [a, b], (a < b), õýð÷èì äýýð òàñðàëòã³é
á°ã°°ä F(x)-ò³³íèé ÿìàð íýãýí ýõ ôóíêö, °°ð°°ð
õýëáýë
∃F (x) = f(x), ∀x ∈ [a, b] ⇐⇒ f(x)dx = F(x) + C
áàéã.
Òîäîðõîéëîëò
[a, b] õýð÷ìýýð f(x) ôóíêöýýñ àâñàí Íüþòîí-Ëåéáíèöèéí òîäîðõîé
èíòåãðàë ãýæ F(x) ôóíêöèéí [a, b] õýð÷èì äýýðõ óòãûí °°ð÷ë°ëò
[F(b)-F(a)] òîîã íýðëýõ áà
b
a
f(x)dx = F(b) − F(a), ýíä F (x) = f(x) (1)
ãýæ òýìäýãëýíý.
x = a öýã íü èíòåãðàëûí äîîä õèë, x = b öýã íü èíòåãðàëûí
äýýä õèë
ÌÀÒÅÌÀÒÈÊ-2

7. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
f(x) ôóíêö [a, b], (a < b), õýð÷èì äýýð òàñðàëòã³é
á°ã°°ä F(x)-ò³³íèé ÿìàð íýãýí ýõ ôóíêö, °°ð°°ð
õýëáýë
∃F (x) = f(x), ∀x ∈ [a, b] ⇐⇒ f(x)dx = F(x) + C
áàéã.
Òîäîðõîéëîëò
[a, b] õýð÷ìýýð f(x) ôóíêöýýñ àâñàí Íüþòîí-Ëåéáíèöèéí òîäîðõîé
èíòåãðàë ãýæ F(x) ôóíêöèéí [a, b] õýð÷èì äýýðõ óòãûí °°ð÷ë°ëò
[F(b)-F(a)] òîîã íýðëýõ áà
b
a
f(x)dx = F(b) − F(a), ýíä F (x) = f(x) (1)
ãýæ òýìäýãëýíý.
x = a öýã íü èíòåãðàëûí äîîä õèë, x = b öýã íü èíòåãðàëûí
äýýä õèë
ÌÀÒÅÌÀÒÈÊ-2

8. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
(1) òýíöýòãýëèéã
b
a
f(x)dx = F(x) |b
a . (∗)
õýëáýðòýé áè÷èæ áîëíî.
Æèøýý
4
1
x2
dx =
x3
3
4
1 =
43
3
−
1
3
=
63
3
π
0
sin xdx = (− cos x) π
0 = − cos(π) + cos 0
= 1+1 = 2
ÌÀÒÅÌÀÒÈÊ-2

9. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
(1) òýíöýòãýëèéã
b
a
f(x)dx = F(x) |b
a . (∗)
õýëáýðòýé áè÷èæ áîëíî.
Æèøýý
4
1
x2
dx =
x3
3
4
1 =
43
3
−
1
3
=
63
3
π
0
sin xdx = (− cos x) π
0 = − cos(π) + cos 0
= 1+1 = 2
ÌÀÒÅÌÀÒÈÊ-2

10. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
(1) òýíöýòãýëèéã
b
a
f(x)dx = F(x) |b
a . (∗)
õýëáýðòýé áè÷èæ áîëíî.
Æèøýý
4
1
x2
dx =
x3
3
4
1 =
43
3
−
1
3
=
63
3
π
0
sin xdx = (− cos x) π
0 = − cos(π) + cos 0
= 1+1 = 2
ÌÀÒÅÌÀÒÈÊ-2

11. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
(1) òýíöýòãýëèéã
b
a
f(x)dx = F(x) |b
a . (∗)
õýëáýðòýé áè÷èæ áîëíî.
Æèøýý
4
1
x2
dx =
x3
3
4
1 =
43
3
−
1
3
=
63
3
π
0
sin xdx = (− cos x) π
0 = − cos(π) + cos 0
= 1+1 = 2
ÌÀÒÅÌÀÒÈÊ-2

12. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
(1) òýíöýòãýëèéã
b
a
f(x)dx = F(x) |b
a . (∗)
õýëáýðòýé áè÷èæ áîëíî.
Æèøýý
4
1
x2
dx =
x3
3
4
1 =
43
3
−
1
3
=
63
3
π
0
sin xdx = (− cos x) π
0 = − cos(π) + cos 0
= 1+1 = 2
ÌÀÒÅÌÀÒÈÊ-2

13. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
(1) òýíöýòãýëèéã
b
a
f(x)dx = F(x) |b
a . (∗)
õýëáýðòýé áè÷èæ áîëíî.
Æèøýý
4
1
x2
dx =
x3
3
4
1 =
43
3
−
1
3
=
63
3
π
0
sin xdx = (− cos x) π
0 = − cos(π) + cos 0
= 1+1 = 2
ÌÀÒÅÌÀÒÈÊ-2

14. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
(1) òýíöýòãýëèéã
b
a
f(x)dx = F(x) |b
a . (∗)
õýëáýðòýé áè÷èæ áîëíî.
Æèøýý
4
1
x2
dx =
x3
3
4
1 =
43
3
−
1
3
=
63
3
π
0
sin xdx = (− cos x) π
0 = − cos(π) + cos 0
= 1+1 = 2
ÌÀÒÅÌÀÒÈÊ-2

15. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
(1) òýíöýòãýëèéã
b
a
f(x)dx = F(x) |b
a . (∗)
õýëáýðòýé áè÷èæ áîëíî.
Æèøýý
4
1
x2
dx =
x3
3
4
1 =
43
3
−
1
3
=
63
3
π
0
sin xdx = (− cos x) π
0 = − cos(π) + cos 0
= 1+1 = 2
ÌÀÒÅÌÀÒÈÊ-2

16. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
(1) òýíöýòãýëèéã
b
a
f(x)dx = F(x) |b
a . (∗)
õýëáýðòýé áè÷èæ áîëíî.
Æèøýý
4
1
x2
dx =
x3
3
4
1 =
43
3
−
1
3
=
63
3
π
0
sin xdx = (− cos x) π
0 = − cos(π) + cos 0
= 1+1 = 2
ÌÀÒÅÌÀÒÈÊ-2

17. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
(1) òýíöýòãýëèéã
b
a
f(x)dx = F(x) |b
a . (∗)
õýëáýðòýé áè÷èæ áîëíî.
Æèøýý
4
1
x2
dx =
x3
3
4
1 =
43
3
−
1
3
=
63
3
π
0
sin xdx = (− cos x) π
0 = − cos(π) + cos 0
= 1+1 = 2
ÌÀÒÅÌÀÒÈÊ-2

18. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Àãóóëãà
1 Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
ÌÀÒÅÌÀÒÈÊ-2

19. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Òîäîðõîéëîëò
[a, b] õýð÷èì äýýð y = f(x)-òàñðàëòã³é ôóíêö °ã°ãäñ°í áà
[a, b] ∀x öýã ò³ð áýõëýí y = f(t) ôóíêöýýñ [a, x] õýð÷ìýýð
àâñàí èíòåãðàë
x
a
f(t)dt
àâ÷ ³çüå. Ýíý èíòåãðàë íü õóâüñàõ äýýä õèëòýé òîäîðõîé
èíòåãðàë ãýæ íýðëýãäýíý. Õýðýâ F(x) íü f(x) ôóíêöèéí ÿìàð
íýãýí ýõ ôóíêö áîë òîäîðõîéëîëò ¼ñîîð
x
a
f(t)dt = F(x) − F(a) (2)
áà ýíä áàéãàà F(x) -äèôôåðåíöèàë÷ëàãäàõ ôóíêö ó÷ðààñ,
îäîî, ýíäýýñ
ÌÀÒÅÌÀÒÈÊ-2

20. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Òîäîðõîéëîëò
ßã ò°ñòýé áàéäëààð, õóâüñàõ äîîä õèëòýé
b
x
f(t)dt, x ∈ [a, b], (x < b),
èíòåãðàëûí òîõèîëäîëä
b
x
f(t)dt = F(b) − F(x) (3)
áîëîõ áà
(∃)
d
dx
b
x
f(t)dt =
d
dt
[F(b) − F(x)] = −F (x) = −f(x)
ÌÀÒÅÌÀÒÈÊ-2

21. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
F(x) =
x
0
√
1 + t4dt, F (x) =?
F (x) =
d
dx
x
0
1 + t4dt = ( 1 + t4) t=x = 1 + x4;
F(x) =
b
x
sin t2dt, F (x) =?
F (x) =
d
dx
b
x
sin t2
dt = [− sin t2
] t=x = − sin x2
.
ÌÀÒÅÌÀÒÈÊ-2

22. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
F(x) =
x
0
√
1 + t4dt, F (x) =?
F (x) =
d
dx
x
0
1 + t4dt = ( 1 + t4) t=x = 1 + x4;
F(x) =
b
x
sin t2dt, F (x) =?
F (x) =
d
dx
b
x
sin t2
dt = [− sin t2
] t=x = − sin x2
.
ÌÀÒÅÌÀÒÈÊ-2

23. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
F(x) =
x
0
√
1 + t4dt, F (x) =?
F (x) =
d
dx
x
0
1 + t4dt = ( 1 + t4) t=x = 1 + x4;
F(x) =
b
x
sin t2dt, F (x) =?
F (x) =
d
dx
b
x
sin t2
dt = [− sin t2
] t=x = − sin x2
.
ÌÀÒÅÌÀÒÈÊ-2

24. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
F(x) =
x
0
√
1 + t4dt, F (x) =?
F (x) =
d
dx
x
0
1 + t4dt = ( 1 + t4) t=x = 1 + x4;
F(x) =
b
x
sin t2dt, F (x) =?
F (x) =
d
dx
b
x
sin t2
dt = [− sin t2
] t=x = − sin x2
.
ÌÀÒÅÌÀÒÈÊ-2

25. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
F(x) =
x
0
√
1 + t4dt, F (x) =?
F (x) =
d
dx
x
0
1 + t4dt = ( 1 + t4) t=x = 1 + x4;
F(x) =
b
x
sin t2dt, F (x) =?
F (x) =
d
dx
b
x
sin t2
dt = [− sin t2
] t=x = − sin x2
.
ÌÀÒÅÌÀÒÈÊ-2

26. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
F(x) =
x
0
√
1 + t4dt, F (x) =?
F (x) =
d
dx
x
0
1 + t4dt = ( 1 + t4) t=x = 1 + x4;
F(x) =
b
x
sin t2dt, F (x) =?
F (x) =
d
dx
b
x
sin t2
dt = [− sin t2
] t=x = − sin x2
.
ÌÀÒÅÌÀÒÈÊ-2

27. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
F(x) =
x
0
√
1 + t4dt, F (x) =?
F (x) =
d
dx
x
0
1 + t4dt = ( 1 + t4) t=x = 1 + x4;
F(x) =
b
x
sin t2dt, F (x) =?
F (x) =
d
dx
b
x
sin t2
dt = [− sin t2
] t=x = − sin x2
.
ÌÀÒÅÌÀÒÈÊ-2

28. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
F(x) =
x
0
√
1 + t4dt, F (x) =?
F (x) =
d
dx
x
0
1 + t4dt = ( 1 + t4) t=x = 1 + x4;
F(x) =
b
x
sin t2dt, F (x) =?
F (x) =
d
dx
b
x
sin t2
dt = [− sin t2
] t=x = − sin x2
.
ÌÀÒÅÌÀÒÈÊ-2

29. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Ñàíàìæ
Õóâüñàõ äýýä (äîîä) õèëòýé òîäîðõîé èíòåãðàëûí õèë íü
°°ð°° ÿìàð íýãýí x-õóâüñàã÷ààñ õàìààðäàã
äèôôåðåíöèàë÷ëàãäàõ ôóíêö áàéõàä
F(x) =
φ(x)
a
f(t)dt
ôóíêöèéí óëàìæëàë íü äàâõàð ôóíêöèéí óëàìæëàëûí
ä³ðìýýð îëäîíî:
d
dx
F(x) =
d
dx
φ(x)
a
f(t)dt =



φ(x)
a
f(t)dt


 · φ (x) =
= f(t) t=φ(x) · φ (x) = f[φ(x)] · φ (x)
ÌÀÒÅÌÀÒÈÊ-2

30. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Ñàíàìæ
ßã èéì áàéäëààð, åð°íõèé í°õö°ëä, f(t)-òàñðàëòã³é,
φ(x), ψ(x)-äèôôåðåíöèàë÷ëàãäàõ ôóíêö³³ä áàéâàë
d
dx
φ(x)
ψ(x)
f(t)dt = f[φ(x)] · φ (x) − f[ψ(x)] · ψ (x).
òîìü¼îãîîð óëàìæëàëûã îëíî.
ÌÀÒÅÌÀÒÈÊ-2

31. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
d
dx
x2
0
√
1 + t2dt =
√
1 + t2
t=x2 · 2x = 2x ·
√
1 + x4;
d
dx
x3
x2
ln tdt, (x > 0), îëîõ
d
dx
x3
x2
ln tdt = (ln t) t=x3 · (x3
) − (ln t)
t=x2
· (x2
) =
= (ln x3
) · 3x2
− (ln x2
)2x = (9x2
− 4x) · ln x.
ÌÀÒÅÌÀÒÈÊ-2

32. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
d
dx
x2
0
√
1 + t2dt =
√
1 + t2
t=x2 · 2x = 2x ·
√
1 + x4;
d
dx
x3
x2
ln tdt, (x > 0), îëîõ
d
dx
x3
x2
ln tdt = (ln t) t=x3 · (x3
) − (ln t)
t=x2
· (x2
) =
= (ln x3
) · 3x2
− (ln x2
)2x = (9x2
− 4x) · ln x.
ÌÀÒÅÌÀÒÈÊ-2

33. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
d
dx
x2
0
√
1 + t2dt =
√
1 + t2
t=x2 · 2x = 2x ·
√
1 + x4;
d
dx
x3
x2
ln tdt, (x > 0), îëîõ
d
dx
x3
x2
ln tdt = (ln t) t=x3 · (x3
) − (ln t)
t=x2
· (x2
) =
= (ln x3
) · 3x2
− (ln x2
)2x = (9x2
− 4x) · ln x.
ÌÀÒÅÌÀÒÈÊ-2

34. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
d
dx
x2
0
√
1 + t2dt =
√
1 + t2
t=x2 · 2x = 2x ·
√
1 + x4;
d
dx
x3
x2
ln tdt, (x > 0), îëîõ
d
dx
x3
x2
ln tdt = (ln t) t=x3 · (x3
) − (ln t)
t=x2
· (x2
) =
= (ln x3
) · 3x2
− (ln x2
)2x = (9x2
− 4x) · ln x.
ÌÀÒÅÌÀÒÈÊ-2

35. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
d
dx
x2
0
√
1 + t2dt =
√
1 + t2
t=x2 · 2x = 2x ·
√
1 + x4;
d
dx
x3
x2
ln tdt, (x > 0), îëîõ
d
dx
x3
x2
ln tdt = (ln t) t=x3 · (x3
) − (ln t)
t=x2
· (x2
) =
= (ln x3
) · 3x2
− (ln x2
)2x = (9x2
− 4x) · ln x.
ÌÀÒÅÌÀÒÈÊ-2

36. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
d
dx
x2
0
√
1 + t2dt =
√
1 + t2
t=x2 · 2x = 2x ·
√
1 + x4;
d
dx
x3
x2
ln tdt, (x > 0), îëîõ
d
dx
x3
x2
ln tdt = (ln t) t=x3 · (x3
) − (ln t)
t=x2
· (x2
) =
= (ln x3
) · 3x2
− (ln x2
)2x = (9x2
− 4x) · ln x.
ÌÀÒÅÌÀÒÈÊ-2

37. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
d
dx
x2
0
√
1 + t2dt =
√
1 + t2
t=x2 · 2x = 2x ·
√
1 + x4;
d
dx
x3
x2
ln tdt, (x > 0), îëîõ
d
dx
x3
x2
ln tdt = (ln t) t=x3 · (x3
) − (ln t)
t=x2
· (x2
) =
= (ln x3
) · 3x2
− (ln x2
)2x = (9x2
− 4x) · ln x.
ÌÀÒÅÌÀÒÈÊ-2

38. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Æèøýý
d
dx
x2
0
√
1 + t2dt =
√
1 + t2
t=x2 · 2x = 2x ·
√
1 + x4;
d
dx
x3
x2
ln tdt, (x > 0), îëîõ
d
dx
x3
x2
ln tdt = (ln t) t=x3 · (x3
) − (ln t)
t=x2
· (x2
) =
= (ln x3
) · 3x2
− (ln x2
)2x = (9x2
− 4x) · ln x.
ÌÀÒÅÌÀÒÈÊ-2

39. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
Àãóóëãà
1 Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
ÌÀÒÅÌÀÒÈÊ-2

40. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
1
b
a
f(x)dx = −
a
b
f(x)dx.
2 Õýðýâ [a, b] õýð÷èì íü x = c öýãýýð [a, c] áà [c, b]
õýð÷ì³³äýä õóâààãäñàí áàéâàë
b
a
f(x)dx =
c
a
f(x)dx +
b
c
f(x)dx
òýíöýòãýë áóþó àääèòèâ ÷àíàð áèåëýãäýíý.
3 Õýðýâ α = const áîë
b
a
α · f(x)dx = α ·
b
a
f(x)dx.
ÌÀÒÅÌÀÒÈÊ-2

41. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
1
b
a
f(x)dx = −
a
b
f(x)dx.
2 Õýðýâ [a, b] õýð÷èì íü x = c öýãýýð [a, c] áà [c, b]
õýð÷ì³³äýä õóâààãäñàí áàéâàë
b
a
f(x)dx =
c
a
f(x)dx +
b
c
f(x)dx
òýíöýòãýë áóþó àääèòèâ ÷àíàð áèåëýãäýíý.
3 Õýðýâ α = const áîë
b
a
α · f(x)dx = α ·
b
a
f(x)dx.
ÌÀÒÅÌÀÒÈÊ-2

42. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
1
b
a
f(x)dx = −
a
b
f(x)dx.
2 Õýðýâ [a, b] õýð÷èì íü x = c öýãýýð [a, c] áà [c, b]
õýð÷ì³³äýä õóâààãäñàí áàéâàë
b
a
f(x)dx =
c
a
f(x)dx +
b
c
f(x)dx
òýíöýòãýë áóþó àääèòèâ ÷àíàð áèåëýãäýíý.
3 Õýðýâ α = const áîë
b
a
α · f(x)dx = α ·
b
a
f(x)dx.
ÌÀÒÅÌÀÒÈÊ-2

43. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
4
b
a
[f(x) ± g(x)]dx =
b
a
f(x)dx ±
b
a
g(x)dx.
Íèéëáýðèéí èíòåãðàë íü èíòåãðàëóóäûí íèéëáýðòýé òýíö³³
áàéíà.
5 Èíòåãðàëûí ìîíîòîí ÷àíàð Õýðýâ f(x) ≥ 0 òàñðàëòã³é
ôóíêö áàéâàë
b
a
f(x)dx ≥ 0, (a < b).
6 Èíòåãðàë áà òýíöýòãýë áèø.
Õýðýâ [a, b] õýð÷èì äýýð f(x), g(x)-òàñðàëòã³é ôóíêöóóä áà
f(x) ≤ g(x), (a ≤ x ≤ b),
òýíöýòãýë áèø áèåëýãäýæ áàéâàë
b
a
f(x)dx ≤
b
a
g(x)dx.
òýíöýòãýë áèø áèåëýãäýíý, °°ð°°ð õýëáýë òýíöýòãýë áèøèéãÌÀÒÅÌÀÒÈÊ-2

44. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
4
b
a
[f(x) ± g(x)]dx =
b
a
f(x)dx ±
b
a
g(x)dx.
Íèéëáýðèéí èíòåãðàë íü èíòåãðàëóóäûí íèéëáýðòýé òýíö³³
áàéíà.
5 Èíòåãðàëûí ìîíîòîí ÷àíàð Õýðýâ f(x) ≥ 0 òàñðàëòã³é
ôóíêö áàéâàë
b
a
f(x)dx ≥ 0, (a < b).
6 Èíòåãðàë áà òýíöýòãýë áèø.
Õýðýâ [a, b] õýð÷èì äýýð f(x), g(x)-òàñðàëòã³é ôóíêöóóä áà
f(x) ≤ g(x), (a ≤ x ≤ b),
òýíöýòãýë áèø áèåëýãäýæ áàéâàë
b
a
f(x)dx ≤
b
a
g(x)dx.
òýíöýòãýë áèø áèåëýãäýíý, °°ð°°ð õýëáýë òýíöýòãýë áèøèéãÌÀÒÅÌÀÒÈÊ-2

45. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
4
b
a
[f(x) ± g(x)]dx =
b
a
f(x)dx ±
b
a
g(x)dx.
Íèéëáýðèéí èíòåãðàë íü èíòåãðàëóóäûí íèéëáýðòýé òýíö³³
áàéíà.
5 Èíòåãðàëûí ìîíîòîí ÷àíàð Õýðýâ f(x) ≥ 0 òàñðàëòã³é
ôóíêö áàéâàë
b
a
f(x)dx ≥ 0, (a < b).
6 Èíòåãðàë áà òýíöýòãýë áèø.
Õýðýâ [a, b] õýð÷èì äýýð f(x), g(x)-òàñðàëòã³é ôóíêöóóä áà
f(x) ≤ g(x), (a ≤ x ≤ b),
òýíöýòãýë áèø áèåëýãäýæ áàéâàë
b
a
f(x)dx ≤
b
a
g(x)dx.
òýíöýòãýë áèø áèåëýãäýíý, °°ð°°ð õýëáýë òýíöýòãýë áèøèéãÌÀÒÅÌÀÒÈÊ-2

46. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
4
1
x2
dx =
x3
3 |4
1
=
43
3
−
1
3
=
63
3
x · ex
dx =
u = x du = dx
dv = ex dx v = ex
= x · ex
− ex
dx
= x · ex
− ex
+ C
= ex
(x − 1) + C
ÌÀÒÅÌÀÒÈÊ-2

47. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
4
1
x2
dx =
x3
3 |4
1
=
43
3
−
1
3
=
63
3
x · ex
dx =
u = x du = dx
dv = ex dx v = ex
= x · ex
− ex
dx
= x · ex
− ex
+ C
= ex
(x − 1) + C
ÌÀÒÅÌÀÒÈÊ-2

48. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
4
1
x2
dx =
x3
3 |4
1
=
43
3
−
1
3
=
63
3
x · ex
dx =
u = x du = dx
dv = ex dx v = ex
= x · ex
− ex
dx
= x · ex
− ex
+ C
= ex
(x − 1) + C
ÌÀÒÅÌÀÒÈÊ-2

49. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
4
1
x2
dx =
x3
3 |4
1
=
43
3
−
1
3
=
63
3
x · ex
dx =
u = x du = dx
dv = ex dx v = ex
= x · ex
− ex
dx
= x · ex
− ex
+ C
= ex
(x − 1) + C
ÌÀÒÅÌÀÒÈÊ-2

50. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
4
1
x2
dx =
x3
3 |4
1
=
43
3
−
1
3
=
63
3
x · ex
dx =
u = x du = dx
dv = ex dx v = ex
= x · ex
− ex
dx
= x · ex
− ex
+ C
= ex
(x − 1) + C
ÌÀÒÅÌÀÒÈÊ-2

51. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
4
1
x2
dx =
x3
3 |4
1
=
43
3
−
1
3
=
63
3
x · ex
dx =
u = x du = dx
dv = ex dx v = ex
= x · ex
− ex
dx
= x · ex
− ex
+ C
= ex
(x − 1) + C
ÌÀÒÅÌÀÒÈÊ-2

52. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
4
1
x2
dx =
x3
3 |4
1
=
43
3
−
1
3
=
63
3
x · ex
dx =
u = x du = dx
dv = ex dx v = ex
= x · ex
− ex
dx
= x · ex
− ex
+ C
= ex
(x − 1) + C
ÌÀÒÅÌÀÒÈÊ-2

53. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
4
1
x2
dx =
x3
3 |4
1
=
43
3
−
1
3
=
63
3
x · ex
dx =
u = x du = dx
dv = ex dx v = ex
= x · ex
− ex
dx
= x · ex
− ex
+ C
= ex
(x − 1) + C
ÌÀÒÅÌÀÒÈÊ-2

54. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
4
1
x2
dx =
x3
3 |4
1
=
43
3
−
1
3
=
63
3
x · ex
dx =
u = x du = dx
dv = ex dx v = ex
= x · ex
− ex
dx
= x · ex
− ex
+ C
= ex
(x − 1) + C
ÌÀÒÅÌÀÒÈÊ-2

55. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
ln xdx =
u = ln x du = dx
x
dv = dx v = x
= x · ln x − x ·
dx
x
= x · ln x − dx = x ln x − x + C
arcsin xdx =
u = arcsin x du = dx√
1−x2
dv = dx v = x
= x · arcsin x −
xdx
√
1 − x2
= x · arcsin x +
1
2
d(1 − x2
)
√
1 − x2
1
2
d(t)
√
t
=
√
t = x · arcsin x + 1 − x2 + C
ÌÀÒÅÌÀÒÈÊ-2

56. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
ln xdx =
u = ln x du = dx
x
dv = dx v = x
= x · ln x − x ·
dx
x
= x · ln x − dx = x ln x − x + C
arcsin xdx =
u = arcsin x du = dx√
1−x2
dv = dx v = x
= x · arcsin x −
xdx
√
1 − x2
= x · arcsin x +
1
2
d(1 − x2
)
√
1 − x2
1
2
d(t)
√
t
=
√
t = x · arcsin x + 1 − x2 + C
ÌÀÒÅÌÀÒÈÊ-2

57. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
ln xdx =
u = ln x du = dx
x
dv = dx v = x
= x · ln x − x ·
dx
x
= x · ln x − dx = x ln x − x + C
arcsin xdx =
u = arcsin x du = dx√
1−x2
dv = dx v = x
= x · arcsin x −
xdx
√
1 − x2
= x · arcsin x +
1
2
d(1 − x2
)
√
1 − x2
1
2
d(t)
√
t
=
√
t = x · arcsin x + 1 − x2 + C
ÌÀÒÅÌÀÒÈÊ-2

58. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
ln xdx =
u = ln x du = dx
x
dv = dx v = x
= x · ln x − x ·
dx
x
= x · ln x − dx = x ln x − x + C
arcsin xdx =
u = arcsin x du = dx√
1−x2
dv = dx v = x
= x · arcsin x −
xdx
√
1 − x2
= x · arcsin x +
1
2
d(1 − x2
)
√
1 − x2
1
2
d(t)
√
t
=
√
t = x · arcsin x + 1 − x2 + C
ÌÀÒÅÌÀÒÈÊ-2

59. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
ln xdx =
u = ln x du = dx
x
dv = dx v = x
= x · ln x − x ·
dx
x
= x · ln x − dx = x ln x − x + C
arcsin xdx =
u = arcsin x du = dx√
1−x2
dv = dx v = x
= x · arcsin x −
xdx
√
1 − x2
= x · arcsin x +
1
2
d(1 − x2
)
√
1 − x2
1
2
d(t)
√
t
=
√
t = x · arcsin x + 1 − x2 + C
ÌÀÒÅÌÀÒÈÊ-2

60. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
ln xdx =
u = ln x du = dx
x
dv = dx v = x
= x · ln x − x ·
dx
x
= x · ln x − dx = x ln x − x + C
arcsin xdx =
u = arcsin x du = dx√
1−x2
dv = dx v = x
= x · arcsin x −
xdx
√
1 − x2
= x · arcsin x +
1
2
d(1 − x2
)
√
1 − x2
1
2
d(t)
√
t
=
√
t = x · arcsin x + 1 − x2 + C
ÌÀÒÅÌÀÒÈÊ-2

61. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
ln xdx =
u = ln x du = dx
x
dv = dx v = x
= x · ln x − x ·
dx
x
= x · ln x − dx = x ln x − x + C
arcsin xdx =
u = arcsin x du = dx√
1−x2
dv = dx v = x
= x · arcsin x −
xdx
√
1 − x2
= x · arcsin x +
1
2
d(1 − x2
)
√
1 − x2
1
2
d(t)
√
t
=
√
t = x · arcsin x + 1 − x2 + C
ÌÀÒÅÌÀÒÈÊ-2

62. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
ln xdx =
u = ln x du = dx
x
dv = dx v = x
= x · ln x − x ·
dx
x
= x · ln x − dx = x ln x − x + C
arcsin xdx =
u = arcsin x du = dx√
1−x2
dv = dx v = x
= x · arcsin x −
xdx
√
1 − x2
= x · arcsin x +
1
2
d(1 − x2
)
√
1 − x2
1
2
d(t)
√
t
=
√
t = x · arcsin x + 1 − x2 + C
ÌÀÒÅÌÀÒÈÊ-2

63. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
ln xdx =
u = ln x du = dx
x
dv = dx v = x
= x · ln x − x ·
dx
x
= x · ln x − dx = x ln x − x + C
arcsin xdx =
u = arcsin x du = dx√
1−x2
dv = dx v = x
= x · arcsin x −
xdx
√
1 − x2
= x · arcsin x +
1
2
d(1 − x2
)
√
1 − x2
1
2
d(t)
√
t
=
√
t = x · arcsin x + 1 − x2 + C
ÌÀÒÅÌÀÒÈÊ-2

64. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
ln xdx =
u = ln x du = dx
x
dv = dx v = x
= x · ln x − x ·
dx
x
= x · ln x − dx = x ln x − x + C
arcsin xdx =
u = arcsin x du = dx√
1−x2
dv = dx v = x
= x · arcsin x −
xdx
√
1 − x2
= x · arcsin x +
1
2
d(1 − x2
)
√
1 − x2
1
2
d(t)
√
t
=
√
t = x · arcsin x + 1 − x2 + C
ÌÀÒÅÌÀÒÈÊ-2

65. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
ln xdx =
u = ln x du = dx
x
dv = dx v = x
= x · ln x − x ·
dx
x
= x · ln x − dx = x ln x − x + C
arcsin xdx =
u = arcsin x du = dx√
1−x2
dv = dx v = x
= x · arcsin x −
xdx
√
1 − x2
= x · arcsin x +
1
2
d(1 − x2
)
√
1 − x2
1
2
d(t)
√
t
=
√
t = x · arcsin x + 1 − x2 + C
ÌÀÒÅÌÀÒÈÊ-2

66. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
ln xdx =
u = ln x du = dx
x
dv = dx v = x
= x · ln x − x ·
dx
x
= x · ln x − dx = x ln x − x + C
arcsin xdx =
u = arcsin x du = dx√
1−x2
dv = dx v = x
= x · arcsin x −
xdx
√
1 − x2
= x · arcsin x +
1
2
d(1 − x2
)
√
1 − x2
1
2
d(t)
√
t
=
√
t = x · arcsin x + 1 − x2 + C
ÌÀÒÅÌÀÒÈÊ-2

67. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
ln xdx =
u = ln x du = dx
x
dv = dx v = x
= x · ln x − x ·
dx
x
= x · ln x − dx = x ln x − x + C
arcsin xdx =
u = arcsin x du = dx√
1−x2
dv = dx v = x
= x · arcsin x −
xdx
√
1 − x2
= x · arcsin x +
1
2
d(1 − x2
)
√
1 − x2
1
2
d(t)
√
t
=
√
t = x · arcsin x + 1 − x2 + C
ÌÀÒÅÌÀÒÈÊ-2

68. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
arctg xdx = u = arctg x du = dx
1+x2
dv = dx v = x
= x · arctg x − xdx√
1+x2
= x · arctg x +
1
2
d(1 + x2
)
√
1 + x2
1
2
d(t)
√
t
=
√
t = x · arctg x + 1 + x2 + C
x2
· cos xdx = u = x2
du = 2xdx
dv = cos xdx v = sin x
= x2
· sin x − 2 x · sin xdx = u = x du = dx
dv = sin xdx v = − cos x
= x2
· sin x − 2( − x cos x + cos xdx)
= x2
· sin x + 2x · cos x − 2 sin x + C)
ÌÀÒÅÌÀÒÈÊ-2

69. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
arctg xdx = u = arctg x du = dx
1+x2
dv = dx v = x
= x · arctg x − xdx√
1+x2
= x · arctg x +
1
2
d(1 + x2
)
√
1 + x2
1
2
d(t)
√
t
=
√
t = x · arctg x + 1 + x2 + C
x2
· cos xdx = u = x2
du = 2xdx
dv = cos xdx v = sin x
= x2
· sin x − 2 x · sin xdx = u = x du = dx
dv = sin xdx v = − cos x
= x2
· sin x − 2( − x cos x + cos xdx)
= x2
· sin x + 2x · cos x − 2 sin x + C)
ÌÀÒÅÌÀÒÈÊ-2

70. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
arctg xdx = u = arctg x du = dx
1+x2
dv = dx v = x
= x · arctg x − xdx√
1+x2
= x · arctg x +
1
2
d(1 + x2
)
√
1 + x2
1
2
d(t)
√
t
=
√
t = x · arctg x + 1 + x2 + C
x2
· cos xdx = u = x2
du = 2xdx
dv = cos xdx v = sin x
= x2
· sin x − 2 x · sin xdx = u = x du = dx
dv = sin xdx v = − cos x
= x2
· sin x − 2( − x cos x + cos xdx)
= x2
· sin x + 2x · cos x − 2 sin x + C)
ÌÀÒÅÌÀÒÈÊ-2

71. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
arctg xdx = u = arctg x du = dx
1+x2
dv = dx v = x
= x · arctg x − xdx√
1+x2
= x · arctg x +
1
2
d(1 + x2
)
√
1 + x2
1
2
d(t)
√
t
=
√
t = x · arctg x + 1 + x2 + C
x2
· cos xdx = u = x2
du = 2xdx
dv = cos xdx v = sin x
= x2
· sin x − 2 x · sin xdx = u = x du = dx
dv = sin xdx v = − cos x
= x2
· sin x − 2( − x cos x + cos xdx)
= x2
· sin x + 2x · cos x − 2 sin x + C)
ÌÀÒÅÌÀÒÈÊ-2

72. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
arctg xdx = u = arctg x du = dx
1+x2
dv = dx v = x
= x · arctg x − xdx√
1+x2
= x · arctg x +
1
2
d(1 + x2
)
√
1 + x2
1
2
d(t)
√
t
=
√
t = x · arctg x + 1 + x2 + C
x2
· cos xdx = u = x2
du = 2xdx
dv = cos xdx v = sin x
= x2
· sin x − 2 x · sin xdx = u = x du = dx
dv = sin xdx v = − cos x
= x2
· sin x − 2( − x cos x + cos xdx)
= x2
· sin x + 2x · cos x − 2 sin x + C)
ÌÀÒÅÌÀÒÈÊ-2

73. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
arctg xdx = u = arctg x du = dx
1+x2
dv = dx v = x
= x · arctg x − xdx√
1+x2
= x · arctg x +
1
2
d(1 + x2
)
√
1 + x2
1
2
d(t)
√
t
=
√
t = x · arctg x + 1 + x2 + C
x2
· cos xdx = u = x2
du = 2xdx
dv = cos xdx v = sin x
= x2
· sin x − 2 x · sin xdx = u = x du = dx
dv = sin xdx v = − cos x
= x2
· sin x − 2( − x cos x + cos xdx)
= x2
· sin x + 2x · cos x − 2 sin x + C)
ÌÀÒÅÌÀÒÈÊ-2

74. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
arctg xdx = u = arctg x du = dx
1+x2
dv = dx v = x
= x · arctg x − xdx√
1+x2
= x · arctg x +
1
2
d(1 + x2
)
√
1 + x2
1
2
d(t)
√
t
=
√
t = x · arctg x + 1 + x2 + C
x2
· cos xdx = u = x2
du = 2xdx
dv = cos xdx v = sin x
= x2
· sin x − 2 x · sin xdx = u = x du = dx
dv = sin xdx v = − cos x
= x2
· sin x − 2( − x cos x + cos xdx)
= x2
· sin x + 2x · cos x − 2 sin x + C)
ÌÀÒÅÌÀÒÈÊ-2

75. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
arctg xdx = u = arctg x du = dx
1+x2
dv = dx v = x
= x · arctg x − xdx√
1+x2
= x · arctg x +
1
2
d(1 + x2
)
√
1 + x2
1
2
d(t)
√
t
=
√
t = x · arctg x + 1 + x2 + C
x2
· cos xdx = u = x2
du = 2xdx
dv = cos xdx v = sin x
= x2
· sin x − 2 x · sin xdx = u = x du = dx
dv = sin xdx v = − cos x
= x2
· sin x − 2( − x cos x + cos xdx)
= x2
· sin x + 2x · cos x − 2 sin x + C)
ÌÀÒÅÌÀÒÈÊ-2

76. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
arctg xdx = u = arctg x du = dx
1+x2
dv = dx v = x
= x · arctg x − xdx√
1+x2
= x · arctg x +
1
2
d(1 + x2
)
√
1 + x2
1
2
d(t)
√
t
=
√
t = x · arctg x + 1 + x2 + C
x2
· cos xdx = u = x2
du = 2xdx
dv = cos xdx v = sin x
= x2
· sin x − 2 x · sin xdx = u = x du = dx
dv = sin xdx v = − cos x
= x2
· sin x − 2( − x cos x + cos xdx)
= x2
· sin x + 2x · cos x − 2 sin x + C)
ÌÀÒÅÌÀÒÈÊ-2

77. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
arctg xdx = u = arctg x du = dx
1+x2
dv = dx v = x
= x · arctg x − xdx√
1+x2
= x · arctg x +
1
2
d(1 + x2
)
√
1 + x2
1
2
d(t)
√
t
=
√
t = x · arctg x + 1 + x2 + C
x2
· cos xdx = u = x2
du = 2xdx
dv = cos xdx v = sin x
= x2
· sin x − 2 x · sin xdx = u = x du = dx
dv = sin xdx v = − cos x
= x2
· sin x − 2( − x cos x + cos xdx)
= x2
· sin x + 2x · cos x − 2 sin x + C)
ÌÀÒÅÌÀÒÈÊ-2

78. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
arctg xdx = u = arctg x du = dx
1+x2
dv = dx v = x
= x · arctg x − xdx√
1+x2
= x · arctg x +
1
2
d(1 + x2
)
√
1 + x2
1
2
d(t)
√
t
=
√
t = x · arctg x + 1 + x2 + C
x2
· cos xdx = u = x2
du = 2xdx
dv = cos xdx v = sin x
= x2
· sin x − 2 x · sin xdx = u = x du = dx
dv = sin xdx v = − cos x
= x2
· sin x − 2( − x cos x + cos xdx)
= x2
· sin x + 2x · cos x − 2 sin x + C)
ÌÀÒÅÌÀÒÈÊ-2

79. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
arctg xdx = u = arctg x du = dx
1+x2
dv = dx v = x
= x · arctg x − xdx√
1+x2
= x · arctg x +
1
2
d(1 + x2
)
√
1 + x2
1
2
d(t)
√
t
=
√
t = x · arctg x + 1 + x2 + C
x2
· cos xdx = u = x2
du = 2xdx
dv = cos xdx v = sin x
= x2
· sin x − 2 x · sin xdx = u = x du = dx
dv = sin xdx v = − cos x
= x2
· sin x − 2( − x cos x + cos xdx)
= x2
· sin x + 2x · cos x − 2 sin x + C)
ÌÀÒÅÌÀÒÈÊ-2

80. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
arctg xdx = u = arctg x du = dx
1+x2
dv = dx v = x
= x · arctg x − xdx√
1+x2
= x · arctg x +
1
2
d(1 + x2
)
√
1 + x2
1
2
d(t)
√
t
=
√
t = x · arctg x + 1 + x2 + C
x2
· cos xdx = u = x2
du = 2xdx
dv = cos xdx v = sin x
= x2
· sin x − 2 x · sin xdx = u = x du = dx
dv = sin xdx v = − cos x
= x2
· sin x − 2( − x cos x + cos xdx)
= x2
· sin x + 2x · cos x − 2 sin x + C)
ÌÀÒÅÌÀÒÈÊ-2

81. Òîäîðõîé èíòåãðàë
Íüþòîí-Ëåéáíèöèéí òîäîðõîé èíòåãðàë
Õóâüñàõ äýýä õèëòýé òîäîðõîé èíòåãðàë
Òîäîðõîé èíòåãðàëûí ³íäñýí ÷àíàðóóä
u · dv = (u · v) − v · du
Æèøýý
arctg xdx = u = arctg x du = dx
1+x2
dv = dx v = x
= x · arctg x − xdx√
1+x2
= x · arctg x +
1
2
d(1 + x2
)
√
1 + x2
1
2
d(t)
√
t
=
√
t = x · arctg x + 1 + x2 + C
x2
· cos xdx = u = x2
du = 2xdx
dv = cos xdx v = sin x
= x2
· sin x − 2 x · sin xdx = u = x du = dx
dv = sin xdx v = − cos x
= x2
· sin x − 2( − x cos x + cos xdx)
= x2
· sin x + 2x · cos x − 2 sin x + C)
ÌÀÒÅÌÀÒÈÊ-2