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Mathematics
Mathematics
Problem
Let
F
(
x
)
=
∫
x
x
2
+
π
6
2
cos
2
t
d
t
F(x) = \int_{x}^{x^2 + \frac{\pi}{6}} 2\cos^2 t \, dt
F
(
x
)
=
∫
x
x
2
+
6
π
2
cos
2
t
d
t
for all
x
∈
R
x \in \mathbb{R}
x
∈
R
and
f
:
[
0
,
1
2
]
→
[
0
,
∞
)
f: \left[0, \frac{1}{2}\right] \to [0, \infty)
f
:
[
0
,
2
1
]
→
[
0
,
∞
)
be a continuous function. For
a
∈
[
0
,
1
2
]
a \in \left[0, \frac{1}{2}\right]
a
∈
[
0
,
2
1
]
, if
F
′
(
a
)
+
2
F'(a) + 2
F
′
(
a
)
+
2
is the area of the region bounded by
x
=
0
x = 0
x
=
0
,
y
=
0
y = 0
y
=
0
,
y
=
f
(
x
)
y = f(x)
y
=
f
(
x
)
and
x
=
a
x = a
x
=
a
, then
f
(
0
)
f(0)
f
(
0
)
is [JEE(Adv.) 2015]
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