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SEEMOUS
2013 SEEMOUS
Problem 1
Problem 1
Part of
2013 SEEMOUS
Problems
(1)
functions satisfing integral equation
Source: SEEMOUS 2013 P1
6/8/2021
Find all continuous functions
f
:
[
1
,
8
]
→
R
f:[1,8]\to\mathbb R
f
:
[
1
,
8
]
→
R
, such that
∫
1
2
f
(
t
3
)
2
d
t
+
2
∫
1
2
f
(
t
3
)
d
t
=
2
3
∫
1
8
f
(
t
)
d
t
−
∫
1
2
(
t
2
−
1
)
2
d
t
.
\int^2_1f(t^3)^2dt+2\int^2_1f(t^3)dt=\frac23\int^8_1f(t)dt-\int^2_1(t^2-1)^2dt.
∫
1
2
f
(
t
3
)
2
d
t
+
2
∫
1
2
f
(
t
3
)
d
t
=
3
2
∫
1
8
f
(
t
)
d
t
−
∫
1
2
(
t
2
−
1
)
2
d
t
.
function
calculus
integration