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Mathematics
Mathematics
Problem
If
I
m
=
∫
(
sin
x
+
cos
x
)
m
d
x
I_m = \int (\sin x + \cos x)^m \, dx
I
m
=
∫
(
sin
x
+
cos
x
)
m
d
x
, then show that
m
I
m
=
(
sin
x
+
cos
x
)
m
−
1
⋅
(
sin
x
−
cos
x
)
+
2
(
m
−
1
)
I
m
−
2
m I_m = (\sin x + \cos x)^{m-1} \cdot (\sin x - \cos x) + 2 (m-1) I_{m-2}
m
I
m
=
(
sin
x
+
cos
x
)
m
−
1
⋅
(
sin
x
−
cos
x
)
+
2
(
m
−
1
)
I
m
−
2
.
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