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Question
Mathematics
If α, β(β>α) are the roots of f(x) ≡ a x2+b x+c=0, a ≠ 0 and f(x) is an even function and I=∫αβ (eff((f(x)/x-α))/f((f(x))x-α)+ef((f(x))x-β), then |I| is equal to
Q. If
α
,
β
(
β
>
α
)
are the roots of
f
(
x
)
≡
a
x
2
+
b
x
+
c
=
0
,
a
=
0
and
f
(
x
)
is an even function and
I
=
∫
α
β
f
(
x
−
α
f
(
x
)
)
+
e
f
(
x
−
β
f
(
x
)
)
e
f
f
(
x
−
α
f
(
x
)
)
, then
∣
I
∣
is equal to
1970
158
Integrals
Report Error
A
∣
∣
b
b
∣
∣
B
∣
2
a
b
C
∣2
a
∣
b
2
−
4
a
c
D
None of these
Solution:
I
=
∫
α
β
e
f
(
x
−
α
a
(
x
−
α
)
(
x
−
β
)
)
e
f
(
x
−
β
a
(
x
−
α
)
(
x
−
β
)
)
e
f
(
x
−
α
a
(
x
−
α
)
(
x
−
β
)
)
d
x
=
∫
α
β
e
f
(
a
(
x
−
β
))
+
f
(
a
(
x
−
α
))
e
f
(
a
(
x
−
β
))
d
x
…
(
1
)
=
∫
α
β
e
f
(
a
(
α
+
β
−
x
−
β
))
+
e
f
(
a
(
α
+
β
−
x
−
α
))
e
f
(
a
(
α
+
β
−
x
−
β
))
d
x
I
=
∫
α
β
e
f
(
a
(
x
−
α
)
+
e
∫
(
a
(
x
−
β
))
e
f
(
a
(
x
−
α
))
d
x
…
(
2
)
2
I
=
∫
α
β
d
x
⇒
I
=
2
∣
α
−
β
∣
=
2∣
a
∣
b
2
−
4
a
c