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Anexo
:
Integrales de funciones inversas trigonométricas
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La siguiente es una lista de
integrales
de
funciones inversas trigonométricas
.
∫
arcsin
x
c
d
x
=
x
arcsin
x
c
+
c
2
−
x
2
{\displaystyle \int \arcsin {\frac {x}{c}}\,dx=x\arcsin {\frac {x}{c}}+{\sqrt {c^{2}-x^{2}}}}
∫
x
arcsin
x
c
d
x
=
(
x
2
2
−
c
2
4
)
arcsin
x
c
+
x
4
c
2
−
x
2
{\displaystyle \int x\arcsin {\frac {x}{c}}\,dx=\left({\frac {x^{2}}{2}}-{\frac {c^{2}}{4}}\right)\arcsin {\frac {x}{c}}+{\frac {x}{4}}{\sqrt {c^{2}-x^{2}}}}
∫
x
2
arcsin
x
c
d
x
=
x
3
3
arcsin
x
c
+
x
2
+
2
c
2
9
c
2
−
x
2
{\displaystyle \int x^{2}\arcsin {\frac {x}{c}}\,dx={\frac {x^{3}}{3}}\arcsin {\frac {x}{c}}+{\frac {x^{2}+2c^{2}}{9}}{\sqrt {c^{2}-x^{2}}}}
∫
arccos
x
c
d
x
=
x
arccos
x
c
−
c
2
−
x
2
{\displaystyle \int \arccos {\frac {x}{c}}\,dx=x\arccos {\frac {x}{c}}-{\sqrt {c^{2}-x^{2}}}}
∫
x
arccos
x
c
d
x
=
(
x
2
2
−
c
2
4
)
arccos
x
c
−
x
4
c
2
−
x
2
{\displaystyle \int x\arccos {\frac {x}{c}}\,dx=\left({\frac {x^{2}}{2}}-{\frac {c^{2}}{4}}\right)\arccos {\frac {x}{c}}-{\frac {x}{4}}{\sqrt {c^{2}-x^{2}}}}
∫
x
2
arccos
x
c
d
x
=
x
3
3
arccos
x
c
−
x
2
+
2
c
2
9
c
2
−
x
2
{\displaystyle \int x^{2}\arccos {\frac {x}{c}}\,dx={\frac {x^{3}}{3}}\arccos {\frac {x}{c}}-{\frac {x^{2}+2c^{2}}{9}}{\sqrt {c^{2}-x^{2}}}}
∫
arctan
x
c
d
x
=
x
arctan
x
c
−
c
2
ln
(
c
2
+
x
2
)
{\displaystyle \int \arctan {\frac {x}{c}}\,dx=x\arctan {\frac {x}{c}}-{\frac {c}{2}}\ln(c^{2}+x^{2})}
∫
x
arctan
x
c
d
x
=
c
2
+
x
2
2
arctan
x
c
−
c
x
2
{\displaystyle \int x\arctan {\frac {x}{c}}\,dx={\frac {c^{2}+x^{2}}{2}}\arctan {\frac {x}{c}}-{\frac {cx}{2}}}
∫
x
2
arctan
x
c
d
x
=
x
3
3
arctan
x
c
−
c
x
2
6
+
c
3
6
ln
c
2
+
x
2
{\displaystyle \int x^{2}\arctan {\frac {x}{c}}\,dx={\frac {x^{3}}{3}}\arctan {\frac {x}{c}}-{\frac {cx^{2}}{6}}+{\frac {c^{3}}{6}}\ln {c^{2}+x^{2}}}
∫
x
n
arctan
x
c
d
x
=
x
n
+
1
n
+
1
arctan
x
c
−
c
n
+
1
∫
x
n
+
1
d
x
c
2
+
x
2
(para
n
≠
−
1
)
{\displaystyle \int x^{n}\arctan {\frac {x}{c}}\,dx={\frac {x^{n+1}}{n+1}}\arctan {\frac {x}{c}}-{\frac {c}{n+1}}\int {\frac {x^{n+1}dx}{c^{2}+x^{2}}}\qquad {\mbox{(para }}n\neq -1{\mbox{)}}}
∫
a
r
c
c
o
t
x
c
d
x
=
x
a
r
c
c
o
t
x
c
+
c
2
ln
(
c
2
+
x
2
)
{\displaystyle \int \mathrm {arccot} \,{\frac {x}{c}}\,dx=x\,\mathrm {arccot} \,{\frac {x}{c}}+{\frac {c}{2}}\ln(c^{2}+x^{2})}
∫
x
a
r
c
c
o
t
x
c
d
x
=
c
2
+
x
2
2
a
r
c
c
o
t
x
c
+
c
x
2
{\displaystyle \int x\,\mathrm {arccot} \,{\frac {x}{c}}\,dx={\frac {c^{2}+x^{2}}{2}}\,\mathrm {arccot} \,{\frac {x}{c}}+{\frac {cx}{2}}}
∫
x
2
a
r
c
c
o
t
x
c
d
x
=
x
3
3
a
r
c
c
o
t
x
c
+
c
x
2
6
−
c
3
6
ln
(
c
2
+
x
2
)
{\displaystyle \int x^{2}\,\mathrm {arccot} \,{\frac {x}{c}}\,dx={\frac {x^{3}}{3}}\,\mathrm {arccot} \,{\frac {x}{c}}+{\frac {cx^{2}}{6}}-{\frac {c^{3}}{6}}\ln(c^{2}+x^{2})}
∫
x
n
a
r
c
c
o
t
x
c
d
x
=
x
n
+
1
n
+
1
a
r
c
c
o
t
x
c
+
c
n
+
1
∫
x
n
+
1
d
x
c
2
+
x
2
(para
n
≠
−
1
)
{\displaystyle \int x^{n}\,\mathrm {arccot} \,{\frac {x}{c}}\,dx={\frac {x^{n+1}}{n+1}}\,\mathrm {arccot} \,{\frac {x}{c}}+{\frac {c}{n+1}}\int {\frac {x^{n+1}dx}{c^{2}+x^{2}}}\qquad {\mbox{(para }}n\neq -1{\mbox{)}}}
