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Mathematics
Mathematics
Problem
If the function
f
:
[
−
1
,
1
]
→
R
f: [-1, 1] \to R
f
:
[
−
1
,
1
]
→
R
is continuous and even, then show that
∫
0
π
/
2
f
(
cos
2
x
)
cos
x
d
x
=
2
∫
0
π
/
4
f
(
sin
2
x
)
cos
x
d
x
.
\int_{0}^{\pi/2} f(\cos 2x) \cos x \, dx = \sqrt{2} \int_{0}^{\pi/4} f(\sin 2x) \cos x \, dx.
∫
0
π
/2
f
(
cos
2
x
)
cos
x
d
x
=
2
∫
0
π
/4
f
(
sin
2
x
)
cos
x
d
x
.
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