f
{\displaystyle f}
F
{\displaystyle F}
f
{\displaystyle f}
f
{\displaystyle f}
F
{\displaystyle F}
En raison de limitations techniques, la typographie souhaitable du titre, «
Exercice : Primitives 4Intégration en mathématiques/Exercices/Primitives 4 », n'a pu être restituée correctement ci-dessus.
f
(
x
)
=
cos
x
ln
(
1
+
cos
x
)
{\displaystyle f(x)=\cos x\ln(1+\cos x)}
f
(
x
)
=
ln
(
x
+
x
2
−
1
)
{\displaystyle f(x)=\ln(x+{\sqrt {x^{2}-1}})}
Solution
f
{\displaystyle f}
[
1
,
+
∞
[
{\displaystyle \left[1,+\infty \right[}
Une intégration par parties donne
∫
f
=
[
x
f
(
x
)
]
−
∫
x
f
′
(
x
)
d
x
=
[
x
f
(
x
)
]
−
∫
x
x
2
−
1
d
x
{\displaystyle \int f=[xf(x)]-\int xf'(x)\,\mathrm {d} x=[xf(x)]-\int {\frac {x}{\sqrt {x^{2}-1}}}\,\mathrm {d} x}
Une primitive de
f
{\displaystyle f}
[
1
,
+
∞
[
{\displaystyle \left[1,+\infty \right[}
F
:
x
↦
x
f
(
x
)
−
x
2
−
1
{\displaystyle F:x\mapsto xf(x)-{\sqrt {x^{2}-1}}}
f
(
x
)
=
x
2
e
x
{\displaystyle f(x)=x^{2}\operatorname {e} ^{x}}
Solution
En enchaînant deux intégrations par parties , on trouve une primitive de
f
{\displaystyle f}
R
{\displaystyle \mathbb {R} }
F
(
x
)
=
(
x
2
−
2
x
+
2
)
e
x
{\displaystyle F(x)=\left(x^{2}-2x+2\right)\operatorname {e} ^{x}}
f
(
x
)
=
e
3
x
{\displaystyle f(x)=\mathrm {e} ^{3x}}
Solution
Une primitive de
f
{\displaystyle f}
R
{\displaystyle \mathbb {R} }
F
=
f
/
3
{\displaystyle F=f/3}
f
(
x
)
=
x
e
−
x
2
{\displaystyle f(x)=x\operatorname {e} ^{-x^{2}}}
Solution
Une primitive de
f
{\displaystyle f}
R
{\displaystyle \mathbb {R} }
F
:
x
↦
−
e
−
x
2
/
2
{\displaystyle F:x\mapsto -\operatorname {e} ^{-x^{2}}/2}
f
(
x
)
=
e
x
1
+
e
x
{\displaystyle f(x)={\frac {\mathrm {e} ^{x}}{1+\mathrm {e} ^{x}}}}
Solution
Une primitive de
f
{\displaystyle f}
R
{\displaystyle \mathbb {R} }
F
:
x
↦
ln
(
1
+
e
x
)
{\displaystyle F:x\mapsto \ln \left(1+\mathrm {e} ^{x}\right)}
f
(
x
)
=
1
x
ln
|
x
|
{\displaystyle f(x)={\frac {1}{x\ln |x|}}}
Solution
Sur chacun des quatre intervalles
]
−
∞
,
−
1
[
{\displaystyle \left]-\infty ,-1\right[}
]
−
1
,
0
[
{\displaystyle \left]-1,0\right[}
]
0
,
1
[
{\displaystyle \left]0,1\right[}
]
1
,
+
∞
[
{\displaystyle \left]1,+\infty \right[}
f
=
u
′
/
u
{\displaystyle f=u'/u}
u
(
x
)
=
ln
|
x
|
{\displaystyle u(x)=\ln |x|}
F
=
ln
|
u
|
{\displaystyle F=\ln |u|}
f
(
x
)
=
cos
(
ln
x
)
x
{\displaystyle f(x)={\frac {\cos(\ln x)}{x}}}
Solution
Une primitive sur
R
+
∗
{\displaystyle \mathbb {R} _{+}^{*}}
f
=
ln
′
×
cos
∘
ln
{\displaystyle f=\ln '\times \cos \circ \ln }
F
=
sin
∘
ln
{\displaystyle F=\sin \circ \ln }
f
(
x
)
=
cos
x
e
sin
x
{\displaystyle f(x)=\cos x\operatorname {e} ^{\sin x}}
Solution
Une primitive de
f
{\displaystyle f}
R
{\displaystyle \mathbb {R} }
F
:
x
↦
e
sin
x
{\displaystyle F:x\mapsto \operatorname {e} ^{\sin x}}
f
(
x
)
=
ln
k
(
x
)
x
{\displaystyle f(x)={\frac {\ln ^{k}(x)}{x}}}
k
∈
N
{\displaystyle k\in \mathbb {N} }
Solution
Une primitive sur
R
+
∗
{\displaystyle \mathbb {R} _{+}^{*}}
f
=
ln
′
ln
k
{\displaystyle f=\ln '\ln ^{k}}
F
=
ln
k
+
1
k
+
1
{\displaystyle F={\frac {\ln ^{k+1}}{k+1}}}
f
(
x
)
=
e
tan
x
1
+
cos
2
x
{\displaystyle f(x)={\frac {\mathrm {e} ^{\tan x}}{1+\cos 2x}}}
f
(
x
)
=
1
sinh
x
=
2
e
x
−
e
−
x
{\displaystyle f(x)={\frac {1}{\sinh x}}={\frac {2}{\mathrm {e} ^{x}-\mathrm {e} ^{-x}}}}
Trigonométrie hyperbolique ).
\pi ,(2k+1)\pi \right[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c47cd2c359f4bd3fd7ed72a3fa8bf84fe4e93f06)
![{\displaystyle \int f=[\sin x\ln(1+\cos x)]+\int {\frac {\sin ^{2}}{1+\cos }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97c8af192e1e76fa2df97cf170bdb0905f680fbc)
![{\displaystyle \int f=[xf(x)]-\int xf'(x)\,\mathrm {d} x=[xf(x)]-\int {\frac {x}{\sqrt {x^{2}-1}}}\,\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6825ce2cca0c03cd42541d11d65825dbb84659b9)
![{\displaystyle \left]0,+\infty \right[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/010ebc9c2d0f3af5ce1c0594c44823f107968561)
![{\displaystyle \left]-\infty ,-1\right[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f241f1d0ad18fa5cb0ac30e10a547354e6e67233)
![{\displaystyle \left]-1,0\right[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/400fa40cdbab6903db5c2d15c60556dfcfd9c83a)
![{\displaystyle \left]0,1\right[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a5471595b2e7b5c1ecc10ea08df3ea3fa5d14a7)
![{\displaystyle \left]1,+\infty \right[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdfacccfcaaf5e33528cd740132906b49a779a81)
![{\displaystyle \left]k\pi -{\frac {\pi }{2}},k\pi +{\frac {\pi }{2}}\right[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed28e882190d9885e60372b85b0896744bfd5fb2)
