I –Өзбетинше жумыс

Funktsiyanin’ aniq emes integrali ha’m oni esaplaw usillari
Meyli f (x) ha’m F(x) funktsiyalari (a,b) da berilgen bolip, F(x) tuwindig’a iye bolsin.
Aniqlama. Eger
F'(x) = f (x) (x e(a,b))
bolsa, onda (a,b) da F(x) funktsiya f (x) funktsiyanin’ da’slepki funktsiyasi delinedi.
Ma’selen,
f (x) = x² (x e(-¥, +¥))
funktsiyanin’ da’ slepki funktsiyasi

(x e(-¥,
3
+¥))
boladi, sebebi
z s ( x³ Y 1 . .
F'(x )= —= — • 3 x ²= x ²= f (x).
I ³ 0 ³
Eger (a,b) da F(x) funktsiya f(x) tin’ da’slepki funktsiyasi bolsa, onda
F(x)+C
barliq da’slepki funktsiyalarinin’ ko’pligi boladi, bunda C qa’legen turaqli san.
Aniqlama. F(x) + C an’latpasi f (x) funktsiyanin’ aniq emes integrali delinedi ha’m j f (x)dx dep belgilenedi:
j f (x) dx = F (x) + C .
Aniq emes integraldin’ tiykarg’i qa’siyetleri:

1. 
   (jf(x)dx)=f(x).
   
2. 
   d(jf (x)dx)=f (x)dx .
   
3. 
   f[ f (x )± g (x )] dx =J f (x) dx ±J g (x) dx .
   
4. 
   jkf(x)dx = kjf(x)dx .
   

Tiykarg’i formulalari

1. 
   jdx =j1 •dx=x+C , bunda C= const.
   

xn+1

2. 
   jxⁿdx = + C (n *-1).
   

n+1

3. 
   j dx = ln II + c .
   

x
a^x

4. 
   ja^xdx = + C (a > 0,a Ф1).
   

lna ^,




























5.

6.

7.

je^xdx=e^x+C.
j sinxdx=-cosx+C. j cos xdx =sin x+C.

8.

dx
. = arcsin x + C.
4\>

9.

dx
I 2 = arctg x + C .
1 + x²

10.

dx
I —~ = - ctgx + C . sin² x

11.

dx
I — = tgx + C.
cos x

12.

13.

J* shxdx = chx + C.
J* chxdx = shx + C.

14.

dx1 x
I ~2 2 = —arctg— + C .
x²+a²aa

15.

dxx
, = arcsin— + C.
a² - x² a

16.

p dx
I 2 2
a-x

= —In
2a

a+x

+C.

17.

dx
x² ± a²

a-x


Tabilg’an integraldin’ durislilig’i tuwindi aliw joli menen tekseriledi.
Endi to’mende integrallawdin’ a’piwayi usillarin keltiremiz:

1. 
   integral astindag’i funktsiyani a’piwayi funktsiyalardin’ qosindisi ko’rinisinde jazip, integraldin’ qa’siyetlerinen paydalaniw usili;
   
2. 
   Differentsial belgisi astina kiritiw usili. Ma’selen,
   

1. 
   dx
   

dx = —d(kx + b), (k, b =const); — = d(ln x); cos xdx = d(sin x); kx

2. 
   
   dx
   = d(tgx), h.t.b.
   cos² x
   

Misallar. To’mendegi aniq emes integrallardi esaplan’:
x⁷

+¥)) 42

9 Эллипс.