Une primitive sur
R
{\displaystyle \mathbb {R} }
f
=
(
1
−
sin
2
)
sin
2
sin
′
{\displaystyle f=(1-\sin ^{2})\sin ^{2}\sin '}
F
=
sin
3
3
−
sin
5
5
{\displaystyle F={\frac {\sin ^{3}}{3}}-{\frac {\sin ^{5}}{5}}}
Une primitive sur
R
{\displaystyle \mathbb {R} }
g
=
(
1
−
sin
2
)
sin
6
sin
′
{\displaystyle g=(1-\sin ^{2})\sin ^{6}\sin '}
G
=
sin
7
7
−
sin
9
9
{\displaystyle G={\frac {\sin ^{7}}{7}}-{\frac {\sin ^{9}}{9}}}
Une primitive sur
R
{\displaystyle \mathbb {R} }
h
=
(
cos
2
−
1
)
cos
4
cos
′
{\displaystyle h=(\cos ^{2}-1)\cos ^{4}\cos '}
H
=
cos
7
7
−
cos
5
5
{\displaystyle H={\frac {\cos ^{7}}{7}}-{\frac {\cos ^{5}}{5}}}
Une primitive sur
R
{\displaystyle \mathbb {R} }
k
=
(
1
−
sin
2
)
2
sin
3
sin
′
{\displaystyle k=(1-\sin ^{2})^{2}\sin ^{3}\sin '}
K
0
=
sin
4
4
−
sin
6
3
+
sin
8
8
{\displaystyle K_{0}={\frac {\sin ^{4}}{4}}-{\frac {\sin ^{6}}{3}}+{\frac {\sin ^{8}}{8}}}
R
{\displaystyle \mathbb {R} }
k
=
(
cos
2
−
1
)
cos
5
cos
′
{\displaystyle k=(\cos ^{2}-1)\cos ^{5}\cos '}
K
1
=
cos
8
8
−
cos
6
6
{\displaystyle K_{1}={\frac {\cos ^{8}}{8}}-{\frac {\cos ^{6}}{6}}}
K
1
=
(
1
−
sin
2
)
4
8
−
(
1
−
sin
2
)
3
6
=
3
(
sin
8
−
4
sin
6
+
6
sin
4
−
4
sin
2
+
1
)
+
4
(
sin
6
−
3
sin
4
+
3
sin
2
−
1
)
24
=
K
0
−
1
24
{\displaystyle K_{1}={\frac {(1-\sin ^{2})^{4}}{8}}-{\frac {(1-\sin ^{2})^{3}}{6}}={\frac {3(\sin ^{8}-4\sin ^{6}+6\sin ^{4}-4\sin ^{2}+1)+4(\sin ^{6}-3\sin ^{4}+3\sin ^{2}-1)}{24}}=K_{0}-{\frac {1}{24}}}
p
{\displaystyle p}
q
{\displaystyle q}
sin
p
cos
q
{\displaystyle \sin ^{p}\cos ^{q}}
u
=
sin
x
{\displaystyle u=\sin x}
q
<
p
{\displaystyle q<p}
v
=
cos
x
{\displaystyle v=\cos x}
p
<
q
{\displaystyle p<q}
Si
p
+
q
{\displaystyle p+q}
w
=
tan
x
{\displaystyle w=\tan x}
cot
x
{\displaystyle \cot x}
Pour
k
{\displaystyle k}
w
=
tan
x
{\displaystyle w=\tan x}
∫
w
3
(
1
+
w
2
)
5
d
w
=
∫
(
w
(
1
+
w
2
)
4
−
w
(
1
+
w
2
)
5
)
d
w
=
1
2
(
(
1
+
w
2
)
−
3
−
3
−
(
1
+
w
2
)
−
4
−
4
)
=
K
1
(
x
)
{\displaystyle \int {\frac {w^{3}}{(1+w^{2})^{5}}}\,\mathrm {d} w=\int \left({\frac {w}{(1+w^{2})^{4}}}-{\frac {w}{(1+w^{2})^{5}}}\right)\,\mathrm {d} w={\frac {1}{2}}\left({\frac {(1+w^{2})^{-3}}{-3}}-{\frac {(1+w^{2})^{-4}}{-4}}\right)=K_{1}(x)}
w
=
cot
x
{\displaystyle w=\cot x}
∫
−
w
5
(
1
+
w
2
)
5
d
w
=
∫
(
−
w
(
1
+
w
2
)
3
+
2
w
(
1
+
w
2
)
4
−
w
(
1
+
w
2
)
5
)
d
w
=
1
2
(
−
(
1
+
w
2
)
−
2
−
2
−
2
(
1
+
w
2
)
−
3
−
3
−
(
1
+
w
2
)
−
4
−
4
)
=
K
0
(
x
)
{\displaystyle \int {\frac {-w^{5}}{(1+w^{2})^{5}}}\,\mathrm {d} w=\int \left({\frac {-w}{(1+w^{2})^{3}}}+{\frac {2w}{(1+w^{2})^{4}}}-{\frac {w}{(1+w^{2})^{5}}}\right)\,\mathrm {d} w={\frac {1}{2}}\left({\frac {-(1+w^{2})^{-2}}{-2}}-{\frac {2(1+w^{2})^{-3}}{-3}}-{\frac {(1+w^{2})^{-4}}{-4}}\right)=K_{0}(x)}
Une primitive sur
R
{\displaystyle \mathbb {R} }
ℓ
:
x
↦
1
8
(
cos
(
4
x
)
−
4
cos
(
2
x
)
+
3
)
{\displaystyle \ell :x\mapsto {\frac {1}{8}}\left(\cos(4x)-4\cos(2x)+3\right)}
L
:
x
↦
1
8
(
sin
(
4
x
)
4
−
2
sin
(
2
x
)
+
3
x
)
{\displaystyle L:x\mapsto {\frac {1}{8}}\left({\frac {\sin(4x)}{4}}-2\sin(2x)+3x\right)}
En posant
w
=
tan
x
{\displaystyle w=\tan x}
∫
w
4
(
1
+
w
2
)
3
d
w
=
∫
(
1
1
+
w
2
−
2
(
1
+
w
2
)
2
+
1
(
1
+
w
2
)
3
)
d
w
{\displaystyle \int {\frac {w^{4}}{(1+w^{2})^{3}}}\,\mathrm {d} w=\int \left({\frac {1}{1+w^{2}}}-{\frac {2}{(1+w^{2})^{2}}}+{\frac {1}{(1+w^{2})^{3}}}\right)\,\mathrm {d} w}
Intégration en mathématiques/Exercices/Primitives 1#Exercice 4-19 )
(
J
1
−
2
J
2
+
J
3
)
(
w
)
=
w
4
(
1
+
w
2
)
2
−
5
w
8
(
1
+
w
2
)
+
3
arctan
w
8
=
sin
x
cos
3
x
4
−
5
sin
x
cos
x
8
+
3
x
8
{\displaystyle (J_{1}-2J_{2}+J_{3})(w)={\frac {w}{4(1+w^{2})^{2}}}-{\frac {5w}{8(1+w^{2})}}+{\frac {3\arctan w}{8}}={\frac {\sin x\cos ^{3}x}{4}}-{\frac {5\sin x\cos x}{8}}+{\frac {3x}{8}}}
sin
(
4
x
)
32
−
sin
(
2
x
)
4
=
sin
x
cos
x
(
2
cos
2
x
−
1
)
8
−
sin
x
cos
x
2
{\displaystyle {\frac {\sin(4x)}{32}}-{\frac {\sin(2x)}{4}}={\frac {\sin x\cos x(2\cos ^{2}x-1)}{8}}-{\frac {\sin x\cos x}{2}}}
sin
x
cos
3
x
4
−
5
sin
x
cos
x
8
{\displaystyle {\frac {\sin x\cos ^{3}x}{4}}-{\frac {5\sin x\cos x}{8}}}
En posant
w
=
cot
x
{\displaystyle w=\cot x}
−
∫
d
w
(
1
+
w
2
)
3
=
−
J
3
(
w
)
=
−
w
4
(
1
+
w
2
)
2
−
3
8
(
w
1
+
w
2
+
arctan
w
)
=
−
3
8
(
π
2
−
x
)
−
3
cos
x
sin
x
8
−
cos
x
sin
3
x
4
{\displaystyle -\int {\frac {\mathrm {d} w}{(1+w^{2})^{3}}}=-J_{3}(w)=-{\frac {w}{4(1+w^{2})^{2}}}-{\frac {3}{8}}\left({\frac {w}{1+w^{2}}}+\arctan w\right)=-{\frac {3}{8}}\left({\frac {\pi }{2}}-x\right)-{\frac {3\cos x\sin x}{8}}-{\frac {\cos x\sin ^{3}x}{4}}}
−
3
cos
x
sin
x
8
−
cos
x
sin
3
x
4
{\displaystyle -{\frac {3\cos x\sin x}{8}}-{\frac {\cos x\sin ^{3}x}{4}}}
sin
x
cos
3
x
4
−
5
sin
x
cos
x
8
{\displaystyle {\frac {\sin x\cos ^{3}x}{4}}-{\frac {5\sin x\cos x}{8}}}
![{\displaystyle I_{k}:=\left]k{\frac {\pi }{2}}+{\frac {\pi }{4}},k{\frac {\pi }{2}}+{\frac {3\pi }{4}}\right[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f73f8572afe2ac65d16b93513841a103e0a340d)
![{\displaystyle \left]k\pi -{\frac {\pi }{2}},k\pi +{\frac {\pi }{2}}\right[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed28e882190d9885e60372b85b0896744bfd5fb2)
![{\displaystyle \left]-\infty ,a\right[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17b52dec7c9e11e0f6ef47e5080c5e1cb036dcad)
![{\displaystyle \left]0,+\infty \right[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/010ebc9c2d0f3af5ce1c0594c44823f107968561)
![{\displaystyle \left[4k\pi ,(4k+2)\pi \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e5699ee35606c6f6ef30f9fcd6762e3498daf1)
![{\displaystyle \left[(4k+2)\pi ,(4k+4)\pi \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca51f7ede385fedb46ac68cae049fd5b21d21614)
