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Mathematics
Mathematics
Problem
Let
f
:
(
0
,
∞
)
→
R
f:(0,\infty)\to\mathbb{R}
f
:
(
0
,
∞
)
→
R
be a continuous function such that
F
(
x
)
=
∫
0
x
t
f
(
t
)
d
t
F(x)=\int_{0}^{x} t f(t)\,dt
F
(
x
)
=
∫
0
x
t
f
(
t
)
d
t
. If
F
(
x
2
)
=
x
4
+
x
5
F(x^2)=x^4+x^5
F
(
x
2
)
=
x
4
+
x
5
, then find
∑
r
=
1
12
f
(
r
2
)
\sum_{r=1}^{12} f(r^2)
∑
r
=
1
12
f
(
r
2
)
.
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