Page 247 - DECO403_MATHEMATICS_FOR_ECONOMISTS_PUNJABI

VED1
E L-LOVELY-H math15-1 IInd 21-10-11 IIIrd 24-1-12 IVth 21-4-12 Vth 20-8-12 Vth 10-9-12

fJekJh^15 L fBPfus nB[ebB

−
= 0 ∫ a f () dt (r[D 1) B'N
t
= 0 ∫ a ( f − ) x dx (r[D 2)

;fEsh IL i/eo f (x), x dk NKe cbB (odd function) j?.
T[d'A f (–x) = – f (x) (gfoGkPk Bkb)

0 a
a ∫ f () dx =− 0 ∫ f () dx
x
x
−
fJ; soQK ;wheoD (i) Bkb,
a a a
a ∫ f () dx =− f () dx + ∫ 0 ∫ f () dx = 0.
x
x
x
− 0
;fEsh IIL i/eo f (x), x dk fi;s cbB (even function) j?.
T[d'A f (–x) = f (x) (gfoGkPk Bkb)
0 a
x
∴ a ∫ f () dx = 0 ∫ f () dx
x
−
fJ; soQK (i) Bkb,
a a a
f () dx = 0 a ∫ f () dx + ∫ 0 ∫ f () dx
x
x
x
−
2= 0 ∫ a f () dx.
x

jZb ;fjs T[dkjoD

T[dkjoD 1L f;ZX eo' fe
À/2 À/2  À 
2
sin x dx = sin 2  − x dx.

∫ 0 ∫ 0  2 

HHHH(1)

HHHH(2)

∴ yZpk gZy & ;Zik gZy.

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