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Mathematics
Mathematics
Problem
Let
I
1
=
∫
0
1
ln
(
1
x
−
1
)
d
x
I_1 = \int_{0}^{1} \ln\left(\frac{1}{x} - 1\right) dx
I
1
=
∫
0
1
ln
(
x
1
−
1
)
d
x
;
I
2
=
∫
0
2
π
(
cos
6
x
)
d
x
I_2 = \int_{0}^{2\pi} (\cos^6 x) dx
I
2
=
∫
0
2
π
(
cos
6
x
)
d
x
;
I
3
=
∫
−
π
2
π
2
(
sin
2
x
)
d
x
I_3 = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\sin^2 x) dx
I
3
=
∫
−
2
π
2
π
(
sin
2
x
)
d
x
, then (1)
I
2
=
I
3
=
0
I_2 = I_3 = 0
I
2
=
I
3
=
0
(2)
I
2
=
I
3
=
0
I_2 = I_3 = 0
I
2
=
I
3
=
0
but
I
1
≠
0
I_1 \neq 0
I
1
=
0
(4)
I
1
=
0
I_1 = 0
I
1
=
0
but
I
3
≠
0
I_3 \neq 0
I
3
=
0
, and
I
2
≠
0
I_2 \neq 0
I
2
=
0
(3)
I
3
=
I
1
=
0
I_3 = I_1 = 0
I
3
=
I
1
=
0
but
I
2
≠
0
I_2 \neq 0
I
2
=
0
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