Integrals

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1. MATHS

FORMULA
INTEGRALS

By

Assistant Professor (Computer Science)
Director, BST, Kokar
&

Assistant Professor (Computer Science)
‘Asst. Director, BSTI, Kokar

IIT-JEE / ENGINEERING

Buddha Science & Technical Institute
K R: 34001. arkhand, India

2. Jharkhand

ERING At Kokar, Ranchi - 834001,

ons

co)

Integration + The inverse process of differentiation is called integration. In

Integral Calculus here are 1x0 types of integral, indefinite integrals and definite

integrals

. INDEFINITE INTEGRALS + The infinite integrals of f(x) is

Jreade= Perec, where LI and € is called constant of

integro. Since Cs pss ay fro des ners are cal indefinite
integras. SI =

Properties of Indefinite Integrals :

© (Roof fra

¡0 LH code ole, NX af el mano

“Formulae

late E

av feu

4c
(ja
(9) findes cos ve C

(VID feosxde=sinxC 7

(vt) rand gt) ¿Cocos x

AIX) foot xdy=logtsin y+

(0%) face = tonte x tn + €
XD Joosecuds = logteosecx-cot29+ €
D face tan see + €

1) foosccveot ue = cosect C

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3. At Kokar, Ranchi - 834001, J

ING

VD [tn tes 0

ov [tasa

avi

D face ade mtn

9 foosecéadr= cote €

5. NOTE à The foma free Zr mei ony vail rabais

Polynom, provide Bae of i exp mb nea

eh

See je

6. Methods or Integration; { Z
‘To fad the integrals, the following eo are use
(0) ‚Integration by decomposition 4
D! Tnegrtion by substitution, including standen integral.
(OU) Integration by partial tions. 2
(av) mtegraton by paris
7. Integration by décomposition :
In his method, he integrand decomposed Se some examples
Mi Jor sige! +sin xx es

A, Pin 2-2)

ay RE tr anise Jaa) cae

Ve

OS

= oe x + cosee nat = fac’ ude foosectude et.

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4. 8. Integration by Substitution including standard integrals:

‘The method in which we change the variable (0 some other variable is called the
method of substitution

When he megan isa the pe Fa) X 1 or LED we pa FC) =1 and
J'ods=dr.
For example 1 je

- RAN
Se

Standard, Integrals: The integrls of he Up _ as

Re EE, | 2
JC Na Fete dx are sand imegral

“ts
arte

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ING

(a) To find be integras | NE Mr OS
{aX r+ persquare and uso the following formulae:
A Sim)

er K
Role ¢
letal
rh

sn (re

= a+

ace a+

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5. At Kokar, Ranchi - 834001, Jharkhand

ING

m Fa NE
DRAN er +c

cm dera

Care

For example:

DEN ee >

(0 To find ints -
a Ha ne armen os

(re

ax
Now by equating the cosicirs of ike terms bot ides nd À And B. Using
rtd of subtiuion and formula of sand

the given integrals.

Express prog

gral 1 1X. we can find

For example:

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6. nula

Or we can use direct method

Jharkhand

Now

ind. ¿CSS method -of substn
section @ =

= pnt à
2 Integration by Partial ration The integral othetynes [+

«dx À 22
FRE tocan be ovale by ring meto pti
aros (es SUPONE

> Traction: If the (degree of num.) (degree of den.), then use long division and
lex the ie ron pe AIO Te prpe ion ca. peca
sm réal Icons af pes =

ERING At Kokar, Ranchi - 834001,

a HE Y
where 4° +bx +Ccánnot be factorized further. À Yr

10, Integration y Part; The oles of he ype | ECON: core found sing

integaon by pata flows
Jr x eo Feo] fate -f[fecode]x rede

Coos and anton coring oe LATE.

NOTE + For the integral fe*[£G0+ dr, split it ito two integral as

Je cou fete and then apply integration by pars in fs integral ony

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ING

11. Definite Integrals We deot he definite integral by | Cu. Geometrical,

J rise ea bunte byte curves y= 60, = bande X-axis

‘There is no need of constant of integration since the definite integral has a unique

value, By fundamental theorem of calculus

Írcom= [Eo FO) Fa where [FCO 100

12. Properties of définit integral The properties of definite integral are very useful
in evalusfing the definite integral moro easily. There are dome properties of
¿ori integral given below: N

6 rom you N
From ya GO
OS = frau HS
mb [riot caos E
Ps: [row [ron o “as
i Py, es ran gags
SN E 0, if f@a-x)= f(x) Ls

e Je ee if f(s evn

0. God

TS pode nnere aces

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8. 13. Definite Integral as the limit ofa sum =

La ref foxde. ln this mebod the area bounded by the cures

E
ls Y= (0, X=4x=b and the X axis i divided into small stripes of equal width
5 h Let the interval [a, 6] is divided imo n equals sub intervals, then
El _( ls

E y N

A a à
2 2. te Fany mistake on his, kindy inform on the mall

Z nal 2074 "BD
El

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