cnc maths1 psi 2015e
TRANSCRIPT
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+0
sin t
t dt
t
1cos t
t2
]0, +[
(, a)
0< < a
a
sin t
t dt=
1cos a
a
1 cos
+
a
1cos t
t2 dt.
+
0
sin t
t
dt
t R \2Z
n N
1
2+
nk=1
cos(kt) = sin 2n+12 t
2sin t2
1
2+
nk=1
cos(kt) t 2Z
t
sin 2n+12 t
2sin t2
0
sin 2n+12 t
2sin t2
dt= 2
g
[0, ]
g(t) =
0
t= 0 ;1
2sin t
2
1t
0< t .
g
C1
[0, ]
0
g(t)sin 2n+12 t dt= 22n+1
0
g(t)cos 2n+12 t dt
0
g(t)sin 2n+12 t dt n+0
0
sin 2n+12 t
t dt
n+
2
+
0
sin t
t
dt=
2
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x
t
1cos(xt)
(1 + t)2
[0, +[
a >0 a
0
sin(xt)
1 + t dt=
1cos(xa)
x(1 + a) +
1
x
a0
1cos(xt)
(1 + t)2 dt.
+0
sin(xt)
1 + t dt
+0
sin(xt)
1 + t dt=
1
x
+0
1cos(xt)
(1 + t)2 dt.
f(x) =
+0
sin(xt)
1 + t dt
x= 0
f
f
+
x >0
0 f(x)
2
x
f +
x >0
f(x) =
1
x
1
x
+0
cos(xt)
(1 + t)2dt
+ f(x) = 1x
+ O ( 1x2
)
f
x >0
f(x) =
+0
1cos u
(x + u)2 du
f
]0, +[
] , 0[
f
]0, +[
h
]0, +[
h(x) =
+0
sin(xt)
(1 + t)3dt
h
h
C1
]0, +[
(x, y)]0, +[2
|h(x) h(y)| 12 |x y|
x >0
f(x) =
1
x
2
x2
+0
sin(xt)
(1 + t)3dt
f
C1
]0, +[
f
x >0
2 f(x) =x
+0
sin u
u(x + u)du
x >0
+0
sin u
u(x + u)du
1 + ln(1 + x)ln x
f 0+
x >0
0< f(x)0f(t)
x > 0
0