Page 246 - DECO403_MATHEMATICS_FOR_ECONOMISTS_PUNJABI

VED1
E L-LOVELY-H math15-1 IInd 6-8-11 IIIrd 24-1-12 IVth 21-4-12 Vth 20-8-12 Vth 10-9-12

noEPk;soh dk rfDs

B'N
∴

b c b
r[D 3H f () dx = f () dx + f () dx, id'Afe a < c < bH
x
x
x
∫ a ∫ a ∫ c

yZpk gZy

;Zik gZy

a a
r[D 4H f x dx = f a − x dx .
)
()
(
∫ 0 ∫ 0
gqwkD (Proof)L wzB fbU a – x = t ∴∴ –dx = dt iK dx = – dt
ns/ id'A x = 0, T[d'A t = a ns/ id'A x = a, T[d'A t = 0
−
−
)
t
t

∴ ;Zik gZy & 0 ∫ ( f a x ) dx = a ∫ 0a f () ( dt =− a ∫ 0 f () dt
= 0 ∫ a f () dt (r[D 1 Bkb)
t
= 0 ∫ a f () dx (r[D 2 Bkb)
x
& yZpk gZy.
a
0
r[D 5H f () dx = , i/eo f (x), x dk Nke cbB (odd function) j?.
x
∫ -a
Gkt f (–x) = – f (x)
a a
a ∫ f () dx = 2 0 ∫ f () dx
x
x
−
i/eo f (x), x dk fi;s cbB (even function) j? Gkt f (–x) = f (x).
a 0 a
gqwkD (Proof)L a ∫ f () dx = a ∫ f () dx + 0 ∫ f () dx ....(i)
x
x
x
− −
∵ – a < 0 < a
0 0
−
−

),
x
t
j[D a ∫ f ( ) dx = a ∫ f ( ) ( dt [x = – t ⇒ dx = – dt oZyD s/]
−
=− a ∫ 0 f () dt
−
−
t

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