8/4/2019 wikimama Class 12 Ch 07 Indefinite Integral

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EHBOMEHERO EHROD^JA

Eh ijh s~jfrefja s~gnaoiv! r`o bo~e}jre}o gm j muhfregh ev de}oh jhb to j~o ~ozue~ob rg mehb r`o

muhfregh t`ef` `jv r`o de}oh bo~e}jre}o R`o s~gfovv gm mehbehd r`ev muhfregh ev fjaaob ehrod~jregh

Bomeheregh< Njvef Fghfosr

Aor M(p% no j bemmo~ohrejnao muhfregh gm p vuf` r`jrb

M p m pbp R`oh M(p% ev fjaaob r`o

ehrod~ja gm m(p% Vingaefjaa! er ev t~erroh jv m(p% bp 9 M(p% m(p%! r`o muhfregh rg no ehrod~jrob! ev

fjaaob r`o ehrod~jhb M(p% ev javg fjaaob r`o jhre)bo~e}jre}o (g~ s~eiere}o muhfregh% gm m(p%

Fghvrjhr gm Ehrod~jreghb (fgr p % fgv of p

bp

>fgv of p bp fgr p f

:

> >

b p :veh

bp j j p

:

> >

: pbp veh f

jj p

8/4/2019 wikimama Class 12 Ch 07 Indefinite Integral

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>

g~:

:

pfgv f

j

:

> >

b p jrjh

bp j p j

:> >

bp : prjh f

j jp j

:>

b :(vof p%bp q p q p :

:>

: bp vof (p% fp p :

:

>

b :(fgvof p%

bp q p q p : :

>

bpfgv of p f

q p q p :

fgvp vehp

fgvp vehp

vehp

rjh p bp bp ah q fgv p q f ah q vof p q ffgvp

vof p(vof p rjh p% p

vof p bp bp ah q vof p rjh p q f ah rjh fvof p rjh p > 2

fgv ofp(fgr p fgv of p% p

fgv ofp bp bp ah (fgr p fgv of p% q f ah rjh ffgr p fgv of p >

9

Vrjhbj~b Mg~iuajo >

> >

bpah p p j f

p j

> >

> >

bpah p p j f

p j

> >

bp : p jah f

>j p jp j

> >

bp : p jagd f

>j p jj p

>> > > > > >p jp j bp p j ah p p j f

> >

>> > > > > >p jp j bp p j ah p p j f

> >

>> > > >p j pj p j p f

> j):bp 9 veh

>

S~gso~reov gm ehbomehero ehrod~jregh

8/4/2019 wikimama Class 12 Ch 07 Indefinite Integral

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eee%b b

m d(p% d(p% bp m d(p%bd bp

e}% Em m(u%bu 9 M(u% + f! r`oh m(jp + n% bp 9:

M jp n f!j

j 3

Ior`gbv gm Ehrod~jregh

Em r`o ehrod~jhb ev hgr j bo~e}jre}o gm j khgth muhfregh! r`oh r`o fg~~ovsghbehd ehrod~jav fjhhgr

no mguhb be~ofra Eh g~bo~ rg mehb r`o ehrod~jav gm fgisaop s~gnaoiv! doho~jaa mgu~ ior`gbv j~o

uvob vg r`jr r`o ehrod~ja ev ~obufob rg r`o vrjhbj~b mg~i

Ehrod~jregh n vunvreruregh g~ n f`jhdo gm r`o ehbosohbohr }j~ejnao

Ehrod~jregh n sj~rv

Ehrod~jregh n uvehd r~edghgior~ef ebohrereov

Ehrod~jregh n sj~reja m~jfreghv

Ehrod~jregh N Vunvreruregh

Be~ofr Vunvrerureghv + j> g~> >p j ! sur p 9 j rjh g~ j fgr

Mg~ ro~iv gm r`o mg~i p> j> g~> >p j ! sur p 9 j vof g~ j fgvof

Mg~ ro~iv gm r`o mg~i j> p> g~ > >j p ! sur p 9 j veh g~ j fgv

Em ngr` j p ! j p j~o s~ovohr! r`oh sur p 9 j fgv

Mg~ r`o rso p j n p ! sur p 9 j fgv> + n veh>

Mg~ r`o rso> > h( p j p% g~ > h( j %>p p ! sur r`o ops~ovvegh ter`eh r`o

n~jfkorv 9 r

Mg~ r`o rso

::: :

h: :h h

>

p n :p j p n g~

p j p j(h H! h 8 :%!

surp n

r

p j

Mg~: >h h

:

p j p n! h:! h> H (jhb 8 :%! jdjeh sur (p + j% 9 r(p + n%

Ehbe~ofr Vunvrerureghv

2>p

p

> >

2 2

p : p :bp bp bp

: p : p :2>

p

>

2 > 2 >

>p >bp! bp

p : kp p : kp

N R~edghgior~ef Rtehv

rjh bp! bpp fgr p !2 2

:bp

veh p fgv p

!

= =

:bp

veh p fgv p!

vehp fgvpbp

j nvehpfgvp

Ehrod~jav gm r~edghgior~ef muhfreghv

m (p%bp veh m(p% f

: m (p%

(eee%m (p%

bp agd m(p% fm(p%

n t~erehd m (p% 9 r

Ehrod~jregh n Sj~rv

Em u jhb } no rtg muhfreghv gm p! r`oh ehrod~ja gm r`o s~gbufr gm r`ovo rtg muhfreghv ev de}oh n

r br

Jr Nr F

(eee%>

:Rg ehrod~jro ! to me~vr sur p 9 :'r

(Jp>

N% Fp B

! vg r`jr

>

> >> > >

:' r br rbr

(J Nr % F BrN% Fp B (J ' r N% F ' r B>bp

(Jp

Hgt r`o vunvreruregh F + Br> 9 u> ~obufov er rg r`o mg~i>j>

bu

u

(e}% Rg o}jaujro

>

>

jp np f bp

bp o mp dp `

to t~ero! jp> +np +f 9 J: (bp +o% ( >mp +d% +N:( bp +o% +F:

t`o~o J:! N: jhb F: j~o fghvrjhrv t`ef` fjh no gnrjehob n fgisj~ehd r`o fgommefeohrv! gm aeko

ro~iv gh ngr` vebov Jhb de}oh ehrod~ja teaa ~obufo rg r`o mg~i

J: : :> > >

>mp d bp bpbp N F

mp dp ` mp dp ` bp o mp dp `

R~edghgior~ef Ehrod~jav

Ehrod~jav gm r`o mg~i ^ (vehp! fgvp% bp

8/4/2019 wikimama Class 12 Ch 07 Indefinite Integral

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s s(e% bp (ee% bp

fgvp + z vehp + ~ fgvp + z vehp

j fgvp+ n vehp + f j fgvp+ n vehp^uao mg~ (e% < Eh r`ev ehrod~ja ops~ovv huio~jrg~ jv a (bohgiehjrg~% + i(bf gm bohgiehjrg~% + hMehb a! i! h n fgisj~ehd r`o fgommefeohrv gm vehp! fgvp jhb fghvrjhr ro~i jhb vsaer r`o ehrod~jaehrg vui gm r`~oo ehrod~jav

bpbp bp hj

bf gm (Bohgiehjrg~%+ iBohgiehjrg~ fgvp + n vehp + f

^uao mg~ (ee% < Ops~ovv huio~jrg~ jv a (bohgiehjrg~% + i(bf gm bohgiehjrg~% jhb mehb a jhb i jvjng}o

Mg~ E 9bp

jfgvp nvehp! to t~ero j 9 ~ fgv ! n 9 ~ veh

jhb dor E 9bp :

vof(p % bp~ fgv(p % ~

9:

agd~

qvof (p % + rjh (p %q + f

t`o~o ~ 9 > >j n jhb 9 rjh :n

j

Ehrod~jregh gm r`o rso< (vehIpfgvHp%bp

I - H hjru~ja huino~v

Em gho gm r`oi ev gbb! r`oh vunvreruro mg~ ro~i gm o}oh sgto~

Em ngr` j~o gbb! vunvreruro oer`o~ gm r`o ro~i

Em ngr` j~o o}oh! uvo r~edghgior~ef ebohrereov gha