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Mathematics
Mathematics
Problem
For the function
f
(
x
)
=
e
2
x
f(x)=e^{2x}
f
(
x
)
=
e
2
x
on
[
0
,
1
]
[0,1]
[
0
,
1
]
, choose the value of
c
∈
(
0
,
1
)
c\in(0,1)
c
∈
(
0
,
1
)
that satisfies
∫
0
1
f
(
x
)
d
x
=
f
(
c
)
(
1
−
0
)
\displaystyle \int_{0}^{1} f(x)\,dx = f(c)\,(1-0)
∫
0
1
f
(
x
)
d
x
=
f
(
c
)
(
1
−
0
)
. a)
ln
(
e
2
−
1
2
)
\ln\!\bigl(\dfrac{e^{2}-1}{2}\bigr)
ln
(
2
e
2
−
1
)
b)
1
2
ln
(
e
2
2
)
\dfrac{1}{2}\ln\!\Bigl(\dfrac{e^{2}}{2}\Bigr)
2
1
ln
(
2
e
2
)
c)
1
2
ln
(
e
2
−
1
2
)
\dfrac{1}{2}\ln\!\bigl(\dfrac{e^{2}-1}{2}\bigr)
2
1
ln
(
2
e
2
−
1
)
d)
1
2
ln
(
e
2
−
1
)
\dfrac{1}{2}\ln(e^{2}-1)
2
1
ln
(
e
2
−
1
)
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