Z Wikiźródeł, wolnej biblioteki
Lista całek nieoznaczonych funkcji niewymiernych
Lista całek nieoznaczonych funkcji niewymiernych
(A01)
∫
a
2
−
x
2
d
x
=
x
2
a
2
−
x
2
+
a
2
2
arcsin
x
|
a
|
(
|
x
|
≤
|
a
|
)
{\displaystyle \qquad \int {\sqrt {a^{2}-x^{2}}}\;dx={\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\arcsin {\frac {x}{|a|}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}
(A02)
∫
x
a
2
−
x
2
d
x
=
−
1
3
(
a
2
−
x
2
)
3
(
|
x
|
≤
|
a
|
)
{\displaystyle \int x{\sqrt {a^{2}-x^{2}}}\;dx=-{\frac {1}{3}}{\sqrt {(a^{2}-x^{2})^{3}}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}
(A03)
∫
x
2
a
2
−
x
2
d
x
=
1
8
a
4
arcsin
x
|
a
|
+
1
8
x
(
2
x
2
−
a
2
)
a
2
−
x
2
(
|
x
|
<
|
a
|
)
{\displaystyle \int x^{2}{\sqrt {a^{2}-x^{2}}}\;dx={\frac {1}{8}}a^{4}\arcsin {\frac {x}{|a|}}+{\frac {1}{8}}x(2x^{2}-a^{2}){\sqrt {a^{2}-x^{2}}}\qquad {\mbox{(}}|x|<|a|{\mbox{)}}}
(A04)
∫
a
2
−
x
2
d
x
x
=
a
2
−
x
2
−
a
ln
|
a
+
a
2
−
x
2
x
|
=
a
2
−
x
2
+
a
ln
|
a
−
a
2
−
x
2
x
|
(
0
<
|
x
|
≤
|
a
|
)
{\displaystyle \int {\frac {{\sqrt {a^{2}-x^{2}}}\;dx}{x}}={\sqrt {a^{2}-x^{2}}}-a\ln \left|{\frac {a+{\sqrt {a^{2}-x^{2}}}}{x}}\right|={\sqrt {a^{2}-x^{2}}}+a\ln \left|{\frac {a-{\sqrt {a^{2}-x^{2}}}}{x}}\right|\qquad {\mbox{(}}0<|x|\leq |a|{\mbox{)}}}
(A05)
∫
a
2
−
x
2
d
x
x
2
=
−
arcsin
x
|
a
|
−
a
2
−
x
2
x
(
0
<
|
x
|
≤
|
a
|
)
{\displaystyle \int {\frac {{\sqrt {a^{2}-x^{2}}}\;dx}{x^{2}}}=-\arcsin {\frac {x}{|a|}}-{\frac {\sqrt {a^{2}-x^{2}}}{x}}\qquad {\mbox{(}}0<|x|\leq |a|{\mbox{)}}}
(A06)
∫
d
x
a
2
−
x
2
=
arcsin
x
|
a
|
(
|
x
|
<
|
a
|
)
{\displaystyle \int {\frac {dx}{\sqrt {a^{2}-x^{2}}}}=\arcsin {\frac {x}{|a|}}\qquad {\mbox{(}}|x|<|a|{\mbox{)}}}
(A07)
∫
x
d
x
a
2
−
x
2
=
−
a
2
−
x
2
(
|
x
|
<
|
a
|
)
{\displaystyle \int {\frac {x\;dx}{\sqrt {a^{2}-x^{2}}}}=-{\sqrt {a^{2}-x^{2}}}\qquad {\mbox{(}}|x|<|a|{\mbox{)}}}
(A08)
∫
x
2
d
x
a
2
−
x
2
=
−
x
2
a
2
−
x
2
+
a
2
2
arcsin
x
|
a
|
(
|
x
|
<
|
a
|
)
{\displaystyle \int {\frac {x^{2}\;dx}{\sqrt {a^{2}-x^{2}}}}=-{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\arcsin {\frac {x}{|a|}}\qquad {\mbox{(}}|x|<|a|{\mbox{)}}}
(A09)
∫
d
x
x
a
2
−
x
2
=
−
1
a
ln
|
a
+
a
2
−
x
2
x
|
=
1
a
ln
|
a
−
a
2
−
x
2
x
|
(
0
<
|
x
|
<
|
a
|
)
{\displaystyle \int {\frac {dx}{x{\sqrt {a^{2}-x^{2}}}}}=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {a^{2}-x^{2}}}}{x}}\right|={\frac {1}{a}}\ln \left|{\frac {a-{\sqrt {a^{2}-x^{2}}}}{x}}\right|\qquad {\mbox{(}}0<|x|<|a|{\mbox{)}}}
(A10)
∫
d
x
x
2
a
2
−
x
2
=
−
a
2
−
x
2
a
2
x
(
0
<
|
x
|
<
|
a
|
)
{\displaystyle \int {\frac {dx}{x^{2}{\sqrt {a^{2}-x^{2}}}}}=-{\frac {\sqrt {a^{2}-x^{2}}}{a^{2}x}}\qquad {\mbox{(}}0<|x|<|a|{\mbox{)}}}
(B01)
∫
x
2
+
a
2
d
x
=
x
2
x
2
+
a
2
+
a
2
2
ln
(
x
+
x
2
+
a
2
)
=
x
2
x
2
+
a
2
−
a
2
2
ln
(
x
2
+
a
2
−
x
)
{\displaystyle \int {\sqrt {x^{2}+a^{2}}}\;dx={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}+{\frac {a^{2}}{2}}\,\ln \left(x+{\sqrt {x^{2}+a^{2}}}\right)={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}-{\frac {a^{2}}{2}}\,\ln \left({\sqrt {x^{2}+a^{2}}}-x\right)}
(B01a)
∫
x
2
+
a
2
d
x
=
x
2
x
2
+
a
2
+
a
2
2
a
r
s
i
n
h
x
|
a
|
{\displaystyle \int {\sqrt {x^{2}+a^{2}}}\;dx={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}+{\frac {a^{2}}{2}}\,\mathrm {arsinh} {\frac {x}{|a|}}}
(B02)
∫
x
x
2
+
a
2
d
x
=
1
3
(
x
2
+
a
2
)
3
{\displaystyle \int x{\sqrt {x^{2}+a^{2}}}\;dx={\frac {1}{3}}{\sqrt {(x^{2}+a^{2})^{3}}}}
(B03)
∫
x
2
x
2
+
a
2
d
x
=
1
8
x
(
2
x
2
+
a
2
)
x
2
+
a
2
−
a
4
8
ln
(
x
+
x
2
+
a
2
)
{\displaystyle \int x^{2}{\sqrt {x^{2}+a^{2}}}\;dx={\frac {1}{8}}\;x(2x^{2}+a^{2}){\sqrt {x^{2}+a^{2}}}-{\frac {a^{4}}{8}}\ln \left(x+{\sqrt {x^{2}+a^{2}}}\right)}
(B04)
∫
x
2
+
a
2
d
x
x
=
x
2
+
a
2
−
a
ln
|
a
+
x
2
+
a
2
x
|
=
x
2
+
a
2
+
a
ln
|
x
2
+
a
2
−
a
x
|
{\displaystyle \int {\frac {{\sqrt {x^{2}+a^{2}}}\;dx}{x}}={\sqrt {x^{2}+a^{2}}}-a\ln \left|{\frac {a+{\sqrt {x^{2}+a^{2}}}}{x}}\right|={\sqrt {x^{2}+a^{2}}}+a\ln \left|{\frac {{\sqrt {x^{2}+a^{2}}}-a}{x}}\right|}
(B05)
∫
x
2
+
a
2
d
x
x
2
=
ln
(
x
+
x
2
+
a
2
)
−
x
2
+
a
2
x
=
−
ln
(
x
2
+
a
2
−
x
)
−
x
2
+
a
2
x
{\displaystyle \int {\frac {{\sqrt {x^{2}+a^{2}}}\;dx}{x^{2}}}=\ln \left(x+{\sqrt {x^{2}+a^{2}}}\right)-{\frac {\sqrt {x^{2}+a^{2}}}{x}}=-\ln \left({\sqrt {x^{2}+a^{2}}}-x\right)-{\frac {\sqrt {x^{2}+a^{2}}}{x}}}
(B06)
∫
d
x
x
2
+
a
2
=
ln
(
x
+
x
2
+
a
2
)
=
−
ln
(
x
2
+
a
2
−
x
)
=
a
r
s
i
n
h
x
|
a
|
{\displaystyle \int {\frac {dx}{\sqrt {x^{2}+a^{2}}}}=\ln \left(x+{\sqrt {x^{2}+a^{2}}}\right)=-\ln \left({\sqrt {x^{2}+a^{2}}}-x\right)=\mathrm {arsinh} {\frac {x}{|a|}}}
(B07)
∫
x
d
x
x
2
+
a
2
=
x
2
+
a
2
{\displaystyle \int {\frac {x\,dx}{\sqrt {x^{2}+a^{2}}}}={\sqrt {x^{2}+a^{2}}}}
(B08)
∫
x
2
d
x
x
2
+
a
2
=
x
2
x
2
+
a
2
−
a
2
2
ln
(
x
+
x
2
+
a
2
)
=
x
2
x
2
+
a
2
+
a
2
2
ln
(
x
2
+
a
2
−
x
)
=
x
2
x
2
+
a
2
−
a
2
2
a
r
s
i
n
h
x
|
a
|
{\displaystyle \int {\frac {x^{2}\;dx}{\sqrt {x^{2}+a^{2}}}}={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}-{\frac {a^{2}}{2}}\ln \left(x+{\sqrt {x^{2}+a^{2}}}\right)={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}+{\frac {a^{2}}{2}}\ln \left({\sqrt {x^{2}+a^{2}}}-x\right)={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}-{\frac {a^{2}}{2}}\,\mathrm {arsinh} {\frac {x}{|a|}}}
(B09)
∫
d
x
x
x
2
+
a
2
=
−
1
a
a
r
s
i
n
h
a
x
=
−
1
a
ln
|
a
+
x
2
+
a
2
x
|
=
1
a
ln
|
x
2
+
a
2
−
a
x
|
{\displaystyle \int {\frac {dx}{x{\sqrt {x^{2}+a^{2}}}}}=-{\frac {1}{a}}\,\mathrm {arsinh} {\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {x^{2}+a^{2}}}}{x}}\right|={\frac {1}{a}}\ln \left|{\frac {{\sqrt {x^{2}+a^{2}}}-a}{x}}\right|}
(B10)
∫
d
x
x
2
x
2
+
a
2
=
−
x
2
+
a
2
a
2
x
{\displaystyle \int {\frac {dx}{x^{2}{\sqrt {x^{2}+a^{2}}}}}=-{\frac {\sqrt {x^{2}+a^{2}}}{a^{2}x}}}
(C01)
∫
x
2
−
a
2
d
x
=
x
2
x
2
−
a
2
−
a
2
2
ln
|
x
+
x
2
−
a
2
|
=
x
2
x
2
−
a
2
+
a
2
2
ln
|
x
−
x
2
−
a
2
|
=
x
2
x
2
−
a
2
−
a
2
2
sgn
x
a
r
c
o
s
h
|
x
a
|
(dla
|
x
|
≥
|
a
|
)
{\displaystyle \int {\sqrt {x^{2}-a^{2}}}\;dx={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}-{\frac {a^{2}}{2}}\ln \left|x+{\sqrt {x^{2}-a^{2}}}\right|={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}+{\frac {a^{2}}{2}}\ln \left|x-{\sqrt {x^{2}-a^{2}}}\right|={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}-{\frac {a^{2}}{2}}\operatorname {sgn} x\,\mathrm {arcosh} \left|{\frac {x}{a}}\right|\qquad {\mbox{(dla }}|x|\geq |a|{\mbox{)}}}
(C02)
∫
x
x
2
−
a
2
d
x
=
1
3
(
x
2
−
a
2
)
3
(dla
|
x
|
≥
|
a
|
)
{\displaystyle \int x{\sqrt {x^{2}-a^{2}}}\;dx={\frac {1}{3}}{\sqrt {(x^{2}-a^{2})^{3}}}\qquad {\mbox{(dla }}|x|\geq |a|{\mbox{)}}}
(C03)
∫
x
2
x
2
−
a
2
d
x
=
1
8
x
(
2
x
2
−
a
2
)
x
2
−
a
2
−
1
8
a
4
ln
|
x
+
x
2
−
a
2
|
(dla
|
x
|
≥
|
a
|
)
{\displaystyle \int x^{2}{\sqrt {x^{2}-a^{2}}}\;dx={\frac {1}{8}}x(2x^{2}-a^{2}){\sqrt {x^{2}-a^{2}}}-{\frac {1}{8}}a^{4}\ln \left|x+{\sqrt {x^{2}-a^{2}}}\right|\qquad {\mbox{(dla }}|x|\geq |a|{\mbox{)}}}
(C04)
∫
x
2
−
a
2
d
x
x
=
x
2
−
a
2
+
a
arcsin
a
|
x
|
=
x
2
−
a
2
−
a
arccos
a
|
x
|
=
x
2
−
a
2
−
a
arctg
x
2
−
a
2
a
(dla
|
x
|
≥
|
a
|
)
{\displaystyle \int {\frac {{\sqrt {x^{2}-a^{2}}}\;dx}{x}}={\sqrt {x^{2}-a^{2}}}+a\arcsin {\frac {a}{|x|}}={\sqrt {x^{2}-a^{2}}}-a\arccos {\frac {a}{|x|}}={\sqrt {x^{2}-a^{2}}}-a\operatorname {arctg} {\frac {\sqrt {x^{2}-a^{2}}}{a}}\qquad {\mbox{(dla }}|x|\geq |a|{\mbox{)}}}
(C05)
∫
x
2
−
a
2
d
x
x
2
=
ln
|
x
+
x
2
−
a
2
|
−
x
2
−
a
2
x
=
−
ln
|
x
−
x
2
−
a
2
|
−
x
2
−
a
2
x
(dla
|
x
|
≥
|
a
|
)
{\displaystyle \int {\frac {{\sqrt {x^{2}-a^{2}}}\;dx}{x^{2}}}=\ln \left|x+{\sqrt {x^{2}-a^{2}}}\right|-{\frac {\sqrt {x^{2}-a^{2}}}{x}}=-\ln \left|x-{\sqrt {x^{2}-a^{2}}}\right|-{\frac {\sqrt {x^{2}-a^{2}}}{x}}\qquad {\mbox{(dla }}|x|\geq |a|{\mbox{)}}}
(C06)
∫
d
x
x
2
−
a
2
=
ln
|
x
+
x
2
−
a
2
|
=
−
ln
|
x
−
x
2
−
a
2
|
=
sgn
x
a
r
c
o
s
h
|
x
a
|
(dla
|
x
|
>
|
a
|
)
{\displaystyle \int {\frac {dx}{\sqrt {x^{2}-a^{2}}}}=\ln \left|x+{\sqrt {x^{2}-a^{2}}}\right|=-\ln \left|x-{\sqrt {x^{2}-a^{2}}}\right|=\operatorname {sgn} x\mathrm {arcosh} \left|{\frac {x}{a}}\right|\qquad {\mbox{(dla }}|x|>|a|{\mbox{)}}}
(C07)
∫
x
d
x
x
2
−
a
2
=
x
2
−
a
2
(dla
|
x
|
>
|
a
|
)
{\displaystyle \int {\frac {x\;dx}{\sqrt {x^{2}-a^{2}}}}={\sqrt {x^{2}-a^{2}}}\qquad {\mbox{(dla }}|x|>|a|{\mbox{)}}}
(C08)
∫
x
2
d
x
x
2
−
a
2
=
x
2
x
2
−
a
2
+
a
2
2
ln
|
x
+
x
2
−
a
2
|
=
x
2
x
2
−
a
2
−
a
2
2
ln
|
x
−
x
2
−
a
2
|
=
x
2
x
2
−
a
2
+
a
2
2
sgn
x
a
r
c
o
s
h
|
x
a
|
(dla
|
x
|
>
|
a
|
)
{\displaystyle \int {\frac {x^{2}\,dx}{\sqrt {x^{2}-a^{2}}}}={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}+{\frac {a^{2}}{2}}\ln \left|x+{\sqrt {x^{2}-a^{2}}}\right|={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}-{\frac {a^{2}}{2}}\ln \left|x-{\sqrt {x^{2}-a^{2}}}\right|={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}+{\frac {a^{2}}{2}}\,\operatorname {sgn} x\,\mathrm {arcosh} \left|{\frac {x}{a}}\right|\qquad {\mbox{(dla }}|x|>|a|{\mbox{)}}}
(C09)
∫
d
x
x
x
2
−
a
2
=
−
1
a
arcsin
a
|
x
|
=
1
a
arccos
a
|
x
|
=
1
a
arctg
x
2
−
a
2
a
(dla
|
x
|
>
|
a
|
)
{\displaystyle \int {\frac {dx}{x{\sqrt {x^{2}-a^{2}}}}}=-{\frac {1}{a}}\,\arcsin {\frac {a}{|x|}}={\frac {1}{a}}\,\arccos {\frac {a}{|x|}}={\frac {1}{a}}\,\operatorname {arctg} {\frac {\sqrt {x^{2}-a^{2}}}{a}}\qquad {\mbox{(dla }}|x|>|a|{\mbox{)}}}
(C10)
∫
d
x
x
2
x
2
−
a
2
=
x
2
−
a
2
a
2
x
(dla
|
x
|
>
|
a
|
)
{\displaystyle \int {\frac {dx}{x^{2}{\sqrt {x^{2}-a^{2}}}}}={\frac {\sqrt {x^{2}-a^{2}}}{a^{2}x}}\qquad {\mbox{(dla }}|x|>|a|{\mbox{)}}}
Następujące wzory są rozwinięciem wzorów (A06), (B06), (C06), dlatego została im nadana numeracja typu (D06x):
(D06a)
∫
d
x
a
x
2
+
b
x
+
c
=
1
a
ln
|
2
a
(
a
x
2
+
b
x
+
c
)
+
2
a
x
+
b
|
(dla
a
>
0
)
{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {a(ax^{2}+bx+c)}}+2ax+b\right|\qquad {\mbox{(dla }}a>0{\mbox{)}}}
(D06b)
∫
d
x
a
x
2
+
b
x
+
c
=
1
a
a
r
s
i
n
h
2
a
x
+
b
4
a
c
−
b
2
(dla
a
>
0
,
4
a
c
−
b
2
>
0
)
{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\,\mathrm {arsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(dla }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}}
(D06c)
∫
d
x
a
x
2
+
b
x
+
c
=
1
a
ln
|
2
a
x
+
b
|
(dla
a
>
0
,
4
a
c
−
b
2
=
0
)
{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\qquad {\mbox{(dla }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}}
(D06d)
∫
d
x
a
x
2
+
b
x
+
c
=
−
1
−
a
arcsin
2
a
x
+
b
b
2
−
4
a
c
(dla
a
<
0
,
4
a
c
−
b
2
<
0
)
{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(dla }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{)}}}
(D06e)
∫
x
d
x
a
x
2
+
b
x
+
c
=
a
x
2
+
b
x
+
c
a
−
b
2
a
∫
d
x
a
x
2
+
b
x
+
c
{\displaystyle \int {\frac {x\;dx}{\sqrt {ax^{2}+bx+c}}}={\frac {\sqrt {ax^{2}+bx+c}}{a}}-{\frac {b}{2a}}\int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}}
Uwaga. Wyrażenia podane jako różne wyniki dla danej całki niekoniecznie muszą być równe - mogą się różnić o pewną stałą (przy czym w poszczególnych przedziałach stałe te mogą być różne).
(E01)
∫
a
x
+
b
n
d
x
=
n
a
(
n
+
1
)
(
a
x
+
b
)
n
+
1
n
+
C
(dla
n
≥
2
)
{\displaystyle \int {\sqrt[{n}]{ax+b}}\;dx={\frac {n}{a\left(n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{n+1}}}+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}}
(E02)
∫
x
a
x
+
b
n
d
x
=
n
a
2
(
2
n
+
1
)
(
a
x
+
b
)
2
n
+
1
n
−
n
b
a
2
(
n
+
1
)
(
a
x
+
b
)
n
+
1
n
+
C
(dla
n
≥
2
)
{\displaystyle \int x{\sqrt[{n}]{ax+b}}\;dx={\frac {n}{a^{2}\left(2n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{2n+1}}}-{\frac {nb}{a^{2}\left(n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{n+1}}}+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}}
(E03)
∫
x
2
a
x
+
b
n
d
x
=
n
a
3
(
3
n
+
1
)
(
a
x
+
b
)
3
n
+
1
n
−
2
n
b
a
3
(
2
n
+
1
)
(
a
x
+
b
)
2
n
+
1
n
+
n
b
2
a
3
(
n
+
1
)
(
a
x
+
b
)
n
+
1
n
+
C
(dla
n
≥
2
)
{\displaystyle \int x^{2}{\sqrt[{n}]{ax+b}}\;dx={\frac {n}{a^{3}\left(3n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{3n+1}}}-{\frac {2nb}{a^{3}\left(2n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{2n+1}}}+{\frac {nb^{2}}{a^{3}\left(n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{n+1}}}+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}}
(E04)
∫
x
m
a
x
+
b
n
d
x
=
n
a
m
+
1
(
∑
k
=
0
m
(
−
1
)
k
(
m
k
)
b
k
n
(
m
−
k
+
1
)
+
1
(
a
x
+
b
)
n
(
m
−
k
+
1
)
+
1
n
)
+
C
(dla
n
≥
2
,
m
≥
0
)
{\displaystyle \int x^{m}{\sqrt[{n}]{ax+b}}\;dx={\frac {n}{a^{m+1}}}\left(\sum _{k=0}^{m}\left(-1\right)^{k}{\frac {{m \choose k}b^{k}}{n\left(m-k+1\right)+1}}{\sqrt[{n}]{\left(ax+b\right)^{n\left(m-k+1\right)+1}}}\right)+C\qquad {\mbox{(dla }}n\geq 2,m\geq 0{\mbox{)}}}
(E05)
∫
d
x
a
x
+
b
n
=
n
a
(
n
−
1
)
(
a
x
+
b
)
n
−
1
n
+
C
(dla
n
≥
2
)
{\displaystyle \int {\frac {dx}{\sqrt[{n}]{ax+b}}}={\frac {n}{a\left(n-1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{n-1}}}+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}}
(E06)
∫
x
d
x
a
x
+
b
n
=
n
a
2
(
1
2
n
−
1
(
a
x
+
b
)
2
n
−
1
n
−
b
n
−
1
(
a
x
+
b
)
n
−
1
n
)
+
C
(dla
n
≥
2
)
{\displaystyle \int {\frac {x\;dx}{\sqrt[{n}]{ax+b}}}={\frac {n}{a^{2}}}\left({\frac {1}{2n-1}}{\sqrt[{n}]{\left(ax+b\right)^{2n-1}}}-{\frac {b}{n-1}}{\sqrt[{n}]{\left(ax+b\right)^{n-1}}}\right)+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}}
(E07)
∫
x
2
d
x
a
x
+
b
n
=
n
a
3
(
1
3
n
−
1
(
a
x
+
b
)
3
n
−
1
n
−
2
b
2
n
−
1
(
a
x
+
b
)
2
n
−
1
n
+
b
2
n
−
1
(
a
x
+
b
)
n
−
1
n
)
+
C
(dla
n
≥
2
)
{\displaystyle \int {\frac {x^{2}\;dx}{\sqrt[{n}]{ax+b}}}={\frac {n}{a^{3}}}\left({\frac {1}{3n-1}}{\sqrt[{n}]{\left(ax+b\right)^{3n-1}}}-{\frac {2b}{2n-1}}{\sqrt[{n}]{\left(ax+b\right)^{2n-1}}}+{\frac {b^{2}}{n-1}}{\sqrt[{n}]{\left(ax+b\right)^{n-1}}}\right)+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}}
(E08)
∫
x
m
d
x
a
x
+
b
n
=
n
a
m
+
1
(
∑
k
=
0
m
(
−
1
)
k
(
m
k
)
b
k
n
(
m
−
k
+
1
)
−
1
(
a
x
+
b
)
n
(
m
−
k
+
1
)
−
1
n
)
+
C
(dla
n
≥
2
,
m
≥
0
)
{\displaystyle \int {\frac {x^{m}\;dx}{\sqrt[{n}]{ax+b}}}={\frac {n}{a^{m+1}}}\left(\sum _{k=0}^{m}\left(-1\right)^{k}{\frac {{m \choose k}b^{k}}{n\left(m-k+1\right)-1}}{\sqrt[{n}]{\left(ax+b\right)^{n\left(m-k+1\right)-1}}}\right)+C\qquad {\mbox{(dla }}n\geq 2,m\geq 0{\mbox{)}}}
![{\displaystyle \int {\sqrt[{n}]{ax+b}}\;dx={\frac {n}{a\left(n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{n+1}}}+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dcc38e73317df0f8ec77d7edba6e057be66ffa9)
![{\displaystyle \int x{\sqrt[{n}]{ax+b}}\;dx={\frac {n}{a^{2}\left(2n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{2n+1}}}-{\frac {nb}{a^{2}\left(n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{n+1}}}+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d7a5c00ec3b5d75fdc0d5e29c2961bf4243089)
![{\displaystyle \int x^{2}{\sqrt[{n}]{ax+b}}\;dx={\frac {n}{a^{3}\left(3n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{3n+1}}}-{\frac {2nb}{a^{3}\left(2n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{2n+1}}}+{\frac {nb^{2}}{a^{3}\left(n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{n+1}}}+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8142a2da327c1e098831e70a834f14d7011dfd)
![{\displaystyle \int x^{m}{\sqrt[{n}]{ax+b}}\;dx={\frac {n}{a^{m+1}}}\left(\sum _{k=0}^{m}\left(-1\right)^{k}{\frac {{m \choose k}b^{k}}{n\left(m-k+1\right)+1}}{\sqrt[{n}]{\left(ax+b\right)^{n\left(m-k+1\right)+1}}}\right)+C\qquad {\mbox{(dla }}n\geq 2,m\geq 0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c553bb60b56fdcaff95ef543647016d14ec4c846)
![{\displaystyle \int {\frac {dx}{\sqrt[{n}]{ax+b}}}={\frac {n}{a\left(n-1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{n-1}}}+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3063a08247a8ea7af008441fab52602732c7f24)
![{\displaystyle \int {\frac {x\;dx}{\sqrt[{n}]{ax+b}}}={\frac {n}{a^{2}}}\left({\frac {1}{2n-1}}{\sqrt[{n}]{\left(ax+b\right)^{2n-1}}}-{\frac {b}{n-1}}{\sqrt[{n}]{\left(ax+b\right)^{n-1}}}\right)+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f8ea72806c41c2514fcaa3ce26f02cd768476a5)
![{\displaystyle \int {\frac {x^{2}\;dx}{\sqrt[{n}]{ax+b}}}={\frac {n}{a^{3}}}\left({\frac {1}{3n-1}}{\sqrt[{n}]{\left(ax+b\right)^{3n-1}}}-{\frac {2b}{2n-1}}{\sqrt[{n}]{\left(ax+b\right)^{2n-1}}}+{\frac {b^{2}}{n-1}}{\sqrt[{n}]{\left(ax+b\right)^{n-1}}}\right)+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4992bf5ba98c7433b396fec91e9893169c9276)
![{\displaystyle \int {\frac {x^{m}\;dx}{\sqrt[{n}]{ax+b}}}={\frac {n}{a^{m+1}}}\left(\sum _{k=0}^{m}\left(-1\right)^{k}{\frac {{m \choose k}b^{k}}{n\left(m-k+1\right)-1}}{\sqrt[{n}]{\left(ax+b\right)^{n\left(m-k+1\right)-1}}}\right)+C\qquad {\mbox{(dla }}n\geq 2,m\geq 0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1440cad64f8ca4820341573ac0daf90ed8b6fbd0)
