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Algâttâs
1
Rationaalfunktioh
2
Eksponentfunktioh
3
Logaritmfunktioh
4
Trigonometrisiih funktioh
5
Arkusfunktioh
6
Hyperbolisiih funktioh
7
Kirjálâšvuotâ
8
Käldeeh
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[
1
]
Rationaalfunktioh
[
mute
|
mute käldee
]
∫
k
d
x
=
k
x
+
C
{\displaystyle \int k\,dx=kx+C}
∫
x
a
d
x
=
x
a
+
1
a
+
1
+
C
(
a
≠
−
1
)
{\displaystyle \int x^{a}\,dx={\frac {x^{a+1}}{a+1}}+C\qquad \ (a\neq -1{\text{)}}\,\!}
∫
(
a
x
+
b
)
n
d
x
=
(
a
x
+
b
)
n
+
1
a
(
n
+
1
)
+
C
(
n
≠
−
1
)
{\displaystyle \int (ax+b)^{n}\,dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad \ (n\neq -1{\text{)}}\,\!}
∫
1
x
d
x
=
ln
|
x
|
+
C
{\displaystyle \int {1 \over x}\,dx=\ln \left|x\right|+C}
∫
c
a
x
+
b
d
x
=
c
a
ln
|
a
x
+
b
|
+
C
{\displaystyle \int {\frac {c}{ax+b}}\,dx={\frac {c}{a}}\ln \left|ax+b\right|+C}
Eksponentfunktioh
[
mute
|
mute käldee
]
∫
e
a
x
d
x
=
1
a
e
a
x
+
C
{\displaystyle \int e^{ax}\,dx={\frac {1}{a}}e^{ax}+C}
∫
f
′
(
x
)
e
f
(
x
)
d
x
=
e
f
(
x
)
+
C
{\displaystyle \int f'(x)e^{f(x)}\,dx=e^{f(x)}+C}
∫
a
x
d
x
=
a
x
ln
a
+
C
{\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln a}}+C}
Logaritmfunktioh
[
mute
|
mute käldee
]
∫
ln
x
d
x
=
x
ln
x
−
x
+
C
{\displaystyle \int \ln x\,dx=x\ln x-x+C}
∫
log
a
x
d
x
=
x
log
a
x
−
x
ln
a
+
C
{\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C}
Trigonometrisiih funktioh
[
mute
|
mute käldee
]
∫
sin
x
d
x
=
−
cos
x
+
C
{\displaystyle \int \sin {x}\,dx=-\cos {x}+C}
∫
cos
x
d
x
=
sin
x
+
C
{\displaystyle \int \cos {x}\,dx=\sin {x}+C}
∫
tan
x
d
x
=
−
ln
|
cos
x
|
+
C
=
ln
|
sec
x
|
+
C
{\displaystyle \int \tan {x}\,dx=-\ln {\left|\cos {x}\right|}+C=\ln {\left|\sec {x}\right|}+C}
∫
cot
x
d
x
=
ln
|
sin
x
|
+
C
{\displaystyle \int \cot {x}\,dx=\ln {\left|\sin {x}\right|}+C}
∫
sec
x
d
x
=
ln
|
sec
x
+
tan
x
|
+
C
{\displaystyle \int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C}
∫
csc
x
d
x
=
ln
|
csc
x
−
cot
x
|
+
C
{\displaystyle \int \csc {x}\,dx=\ln {\left|\csc {x}-\cot {x}\right|}+C}
∫
sec
2
x
d
x
=
tan
x
+
C
{\displaystyle \int \sec ^{2}x\,dx=\tan x+C}
∫
csc
2
x
d
x
=
−
cot
x
+
C
{\displaystyle \int \csc ^{2}x\,dx=-\cot x+C}
∫
sec
x
tan
x
d
x
=
sec
x
+
C
{\displaystyle \int \sec {x}\,\tan {x}\,dx=\sec {x}+C}
∫
csc
x
cot
x
d
x
=
−
csc
x
+
C
{\displaystyle \int \csc {x}\,\cot {x}\,dx=-\csc {x}+C}
∫
sin
2
x
d
x
=
1
2
(
x
−
sin
2
x
2
)
+
C
=
1
2
(
x
−
sin
x
cos
x
)
+
C
{\displaystyle \int \sin ^{2}x\,dx={\frac {1}{2}}\left(x-{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x-\sin x\cos x)+C}
∫
cos
2
x
d
x
=
1
2
(
x
+
sin
2
x
2
)
+
C
=
1
2
(
x
+
sin
x
cos
x
)
+
C
{\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}}\left(x+{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x+\sin x\cos x)+C}
∫
sec
3
x
d
x
=
1
2
sec
x
tan
x
+
1
2
ln
|
sec
x
+
tan
x
|
+
C
{\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C}
∫
sin
n
x
d
x
=
−
sin
n
−
1
x
cos
x
n
+
n
−
1
n
∫
sin
n
−
2
x
d
x
{\displaystyle \int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx}
∫
cos
n
x
d
x
=
cos
n
−
1
x
sin
x
n
+
n
−
1
n
∫
cos
n
−
2
x
d
x
{\displaystyle \int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx}
Arkusfunktioh
[
mute
|
mute käldee
]
∫
arcsin
x
d
x
=
x
arcsin
x
+
1
−
x
2
+
C
,
|
x
|
≤
+
1
{\displaystyle \int \arcsin {x}\,dx=x\arcsin {x}+{\sqrt {1-x^{2}}}+C,\vert x\vert \leq +1}
∫
arccos
x
d
x
=
x
arccos
x
−
1
−
x
2
+
C
,
|
x
|
≤
+
1
{\displaystyle \int \arccos {x}\,dx=x\arccos {x}-{\sqrt {1-x^{2}}}+C,\vert x\vert \leq +1}
∫
arctan
x
d
x
=
x
arctan
x
−
1
2
ln
|
1
+
x
2
|
+
C
,
{\displaystyle \int \arctan {x}\,dx=x\arctan {x}-{\frac {1}{2}}\ln {\vert 1+x^{2}\vert }+C,}
∫
arccot
x
d
x
=
x
arccot
x
+
1
2
ln
|
1
+
x
2
|
+
C
,
{\displaystyle \int \operatorname {arccot} {x}\,dx=x\operatorname {arccot} {x}+{\frac {1}{2}}\ln {\vert 1+x^{2}\vert }+C,}
∫
arcsec
x
d
x
=
x
arcsec
x
−
ln
|
x
(
1
+
1
−
x
−
2
)
|
+
C
,
|
x
|
≥
+
1
{\displaystyle \int \operatorname {arcsec} {x}\,dx=x\operatorname {arcsec} {x}-\ln \vert x\,(1+{\sqrt {1-x^{-2}}}\,)\vert +C,\vert x\vert \geq +1}
∫
arccsc
x
d
x
=
x
arccsc
x
+
ln
|
x
(
1
+
1
−
x
−
2
)
|
+
C
,
|
x
|
≥
+
1
{\displaystyle \int \operatorname {arccsc} {x}\,dx=x\operatorname {arccsc} {x}+\ln \vert x\,(1+{\sqrt {1-x^{-2}}}\,)\vert +C,\vert x\vert \geq +1}
Hyperbolisiih funktioh
[
mute
|
mute käldee
]
∫
sinh
x
d
x
=
cosh
x
+
C
{\displaystyle \int \sinh x\,dx=\cosh x+C}
∫
cosh
x
d
x
=
sinh
x
+
C
{\displaystyle \int \cosh x\,dx=\sinh x+C}
∫
tanh
x
d
x
=
ln
cosh
x
+
C
{\displaystyle \int \tanh x\,dx=\ln \cosh x+C}
∫
coth
x
d
x
=
ln
|
sinh
x
|
+
C
,
x
≠
0
{\displaystyle \int \coth x\,dx=\ln |\sinh x|+C,x\neq 0}
∫
sech
x
d
x
=
arctan
(
sinh
x
)
+
C
{\displaystyle \int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C}
∫
csch
x
d
x
=
ln
|
tanh
x
2
|
+
C
,
x
≠
0
{\displaystyle \int \operatorname {csch} \,x\,dx=\ln \left|\tanh {x \over 2}\right|+C,x\neq 0}
∫
d
x
sinh
a
x
=
1
a
ln
|
tanh
a
x
2
|
+
C
{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|\tanh {\frac {ax}{2}}\right|+C\,}
∫
d
x
cosh
a
x
=
2
a
arctan
e
a
x
+
C
{\displaystyle \int {\frac {dx}{\cosh ax}}={\frac {2}{a}}\arctan e^{ax}+C\,}
Kirjálâšvuotâ
[
mute
|
mute käldee
]
Spiegel, Murray M.:
Mathematical Handbook of Formulas and Tables
. Shaum's Outline Series. McGraw-Hill, 1992.
ISBN 0-07-060224-7
.
Käldeeh
[
mute
|
mute käldee
]
↑
Valtanen, Esko:
Matemaattisia kaavoja ja taulukoita
, s. 78–80. , 2013.
ISBN 978-952-9867-37-0
.
(suomâkielân)
Jurgâlus
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