- Source: Daftar integral dari fungsi logaritmik
Berikut daftar integral dari fungsi logaritmik. Untuk daftar integral lainnya, lihat tabel integral.
Integral hanya melibatkan fungsi logaritmik
(dengan asumsi
x
>
0
{\displaystyle x>0}
, dan konstanta integrasi tidak diperlihatkankan)
∫
ln
c
x
d
x
=
x
ln
c
x
−
x
{\displaystyle \int \ln cx\;dx=x\ln cx-x}
∫
ln
(
a
x
+
b
)
d
x
=
x
ln
(
a
x
+
b
)
−
x
+
b
a
ln
(
a
x
+
b
)
{\displaystyle \int \ln(ax+b)\;dx=x\ln(ax+b)-x+{\frac {b}{a}}\ln(ax+b)}
∫
(
ln
x
)
2
d
x
=
x
(
ln
x
)
2
−
2
x
ln
x
+
2
x
{\displaystyle \int (\ln x)^{2}\;dx=x(\ln x)^{2}-2x\ln x+2x}
∫
(
ln
c
x
)
n
d
x
=
x
(
ln
c
x
)
n
−
n
∫
(
ln
c
x
)
n
−
1
d
x
{\displaystyle \int (\ln cx)^{n}\;dx=x(\ln cx)^{n}-n\int (\ln cx)^{n-1}dx}
∫
d
x
ln
x
=
ln
|
ln
x
|
+
ln
x
+
∑
i
=
2
∞
(
ln
x
)
i
i
⋅
i
!
{\displaystyle \int {\frac {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum _{i=2}^{\infty }{\frac {(\ln x)^{i}}{i\cdot i!}}}
∫
d
x
(
ln
x
)
n
=
−
x
(
n
−
1
)
(
ln
x
)
n
−
1
+
1
n
−
1
∫
d
x
(
ln
x
)
n
−
1
{\displaystyle \int {\frac {dx}{(\ln x)^{n}}}=-{\frac {x}{(n-1)(\ln x)^{n-1}}}+{\frac {1}{n-1}}\int {\frac {dx}{(\ln x)^{n-1}}}}
untuk
n
≠
1
{\displaystyle n\neq 1}
∫
x
m
ln
x
d
x
=
x
m
+
1
(
ln
x
m
+
1
−
1
(
m
+
1
)
2
)
{\displaystyle \int x^{m}\ln x\;dx=x^{m+1}\left({\frac {\ln x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)}
untuk
m
≠
−
1
{\displaystyle m\neq -1}
∫
x
m
(
ln
x
)
n
d
x
=
x
m
+
1
(
ln
x
)
n
m
+
1
−
n
m
+
1
∫
x
m
(
ln
x
)
n
−
1
d
x
{\displaystyle \int x^{m}(\ln x)^{n}\;dx={\frac {x^{m+1}(\ln x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m}(\ln x)^{n-1}dx}
untuk
m
≠
−
1
{\displaystyle m\neq -1}
∫
(
ln
x
)
n
d
x
x
=
(
ln
x
)
n
+
1
n
+
1
{\displaystyle \int {\frac {(\ln x)^{n}\;dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}}
untuk
n
≠
−
1
{\displaystyle n\neq -1}
∫
ln
x
n
d
x
x
=
(
ln
x
n
)
2
2
n
{\displaystyle \int {\frac {\ln {x^{n}}\;dx}{x}}={\frac {(\ln {x^{n}})^{2}}{2n}}}
untuk
n
≠
0
{\displaystyle n\neq 0}
∫
ln
x
d
x
x
m
=
−
ln
x
(
m
−
1
)
x
m
−
1
−
1
(
m
−
1
)
2
x
m
−
1
{\displaystyle \int {\frac {\ln x\,dx}{x^{m}}}=-{\frac {\ln x}{(m-1)x^{m-1}}}-{\frac {1}{(m-1)^{2}x^{m-1}}}}
untuk
m
≠
1
{\displaystyle m\neq 1}
∫
(
ln
x
)
n
d
x
x
m
=
−
(
ln
x
)
n
(
m
−
1
)
x
m
−
1
+
n
m
−
1
∫
(
ln
x
)
n
−
1
d
x
x
m
{\displaystyle \int {\frac {(\ln x)^{n}\;dx}{x^{m}}}=-{\frac {(\ln x)^{n}}{(m-1)x^{m-1}}}+{\frac {n}{m-1}}\int {\frac {(\ln x)^{n-1}dx}{x^{m}}}}
untuk
m
≠
1
{\displaystyle m\neq 1}
∫
x
m
d
x
(
ln
x
)
n
=
−
x
m
+
1
(
n
−
1
)
(
ln
x
)
n
−
1
+
m
+
1
n
−
1
∫
x
m
d
x
(
ln
x
)
n
−
1
{\displaystyle \int {\frac {x^{m}\;dx}{(\ln x)^{n}}}=-{\frac {x^{m+1}}{(n-1)(\ln x)^{n-1}}}+{\frac {m+1}{n-1}}\int {\frac {x^{m}dx}{(\ln x)^{n-1}}}}
untuk
n
≠
1
{\displaystyle n\neq 1}
∫
d
x
x
ln
x
=
ln
|
ln
x
|
{\displaystyle \int {\frac {dx}{x\ln x}}=\ln \left|\ln x\right|}
∫
d
x
x
n
ln
x
=
ln
|
ln
x
|
+
∑
i
=
1
∞
(
−
1
)
i
(
n
−
1
)
i
(
ln
x
)
i
i
⋅
i
!
{\displaystyle \int {\frac {dx}{x^{n}\ln x}}=\ln \left|\ln x\right|+\sum _{i=1}^{\infty }(-1)^{i}{\frac {(n-1)^{i}(\ln x)^{i}}{i\cdot i!}}}
∫
d
x
x
(
ln
x
)
n
=
−
1
(
n
−
1
)
(
ln
x
)
n
−
1
{\displaystyle \int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}}
untuk
n
≠
1
{\displaystyle n\neq 1}
∫
ln
(
x
2
+
a
2
)
d
x
=
x
ln
(
x
2
+
a
2
)
−
2
x
+
2
a
tan
−
1
x
a
{\displaystyle \int \ln(x^{2}+a^{2})\;dx=x\ln(x^{2}+a^{2})-2x+2a\tan ^{-1}{\frac {x}{a}}}
∫
x
x
2
+
a
2
ln
(
x
2
+
a
2
)
d
x
=
1
4
ln
2
(
x
2
+
a
2
)
{\displaystyle \int {\frac {x}{x^{2}+a^{2}}}\ln(x^{2}+a^{2})\;dx={\frac {1}{4}}\ln ^{2}(x^{2}+a^{2})}
∫
sin
(
ln
x
)
d
x
=
x
2
(
sin
(
ln
x
)
−
cos
(
ln
x
)
)
{\displaystyle \int \sin(\ln x)\;dx={\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))}
∫
cos
(
ln
x
)
d
x
=
x
2
(
sin
(
ln
x
)
+
cos
(
ln
x
)
)
{\displaystyle \int \cos(\ln x)\;dx={\frac {x}{2}}(\sin(\ln x)+\cos(\ln x))}
∫
e
x
(
x
ln
x
−
x
−
1
x
)
d
x
=
e
x
(
x
ln
x
−
x
−
ln
x
)
{\displaystyle \int e^{x}\left(x\ln x-x-{\frac {1}{x}}\right)\;dx=e^{x}(x\ln x-x-\ln x)}
∫
1
e
x
(
1
x
−
ln
x
)
d
x
=
ln
x
e
x
{\displaystyle \int {\frac {1}{e^{x}}}\left({\frac {1}{x}}-\ln x\right)\;dx={\frac {\ln x}{e^{x}}}}
∫
e
x
(
1
ln
x
−
1
x
ln
2
x
)
d
x
=
e
x
ln
x
{\displaystyle \int e^{x}\left({\frac {1}{\ln x}}-{\frac {1}{x\ln ^{2}x}}\right)\;dx={\frac {e^{x}}{\ln x}}}
Integral yang melibatkan fungsi logaritmik dan pangkat
(dengan asumsi
x
>
0
{\displaystyle x>0}
, dan konstanta integrasi tidak diperlihatkankan)
∫
x
m
ln
x
d
x
=
x
m
+
1
(
ln
x
m
+
1
−
1
(
m
+
1
)
2
)
{\displaystyle \int x^{m}\ln x\,dx=x^{m+1}\left({\frac {\ln x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)}
untuk
m
≠
−
1
{\displaystyle m\neq -1}
∫
x
m
(
ln
x
)
n
d
x
=
x
m
+
1
(
ln
x
)
n
m
+
1
−
n
m
+
1
∫
x
m
(
ln
x
)
n
−
1
d
x
{\displaystyle \int x^{m}(\ln x)^{n}\,dx={\frac {x^{m+1}(\ln x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m}(\ln x)^{n-1}dx}
untuk untuk
m
≠
−
1
{\displaystyle m\neq -1}
∫
(
ln
x
)
n
d
x
x
=
(
ln
x
)
n
+
1
n
+
1
{\displaystyle \int {\frac {(\ln x)^{n}\,dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}}
untuk
n
≠
−
1
{\displaystyle n\neq -1}
∫
ln
x
d
x
x
m
=
−
ln
x
(
m
−
1
)
x
m
−
1
−
1
(
m
−
1
)
2
x
m
−
1
{\displaystyle \int {\frac {\ln x\,dx}{x^{m}}}=-{\frac {\ln x}{(m-1)x^{m-1}}}-{\frac {1}{(m-1)^{2}x^{m-1}}}}
untuk
m
≠
1
{\displaystyle m\neq 1}
∫
(
ln
x
)
n
d
x
x
m
=
−
(
ln
x
)
n
(
m
−
1
)
x
m
−
1
+
n
m
−
1
∫
(
ln
x
)
n
−
1
d
x
x
m
{\displaystyle \int {\frac {(\ln x)^{n}\,dx}{x^{m}}}=-{\frac {(\ln x)^{n}}{(m-1)x^{m-1}}}+{\frac {n}{m-1}}\int {\frac {(\ln x)^{n-1}dx}{x^{m}}}}
untuk
m
≠
1
{\displaystyle m\neq 1}
∫
x
m
d
x
(
ln
x
)
n
=
−
x
m
+
1
(
n
−
1
)
(
ln
x
)
n
−
1
+
m
+
1
n
−
1
∫
x
m
d
x
(
ln
x
)
n
−
1
{\displaystyle \int {\frac {x^{m}\,dx}{(\ln x)^{n}}}=-{\frac {x^{m+1}}{(n-1)(\ln x)^{n-1}}}+{\frac {m+1}{n-1}}\int {\frac {x^{m}dx}{(\ln x)^{n-1}}}}
untuk
n
≠
1
{\displaystyle n\neq 1}
∫
d
x
x
ln
x
=
ln
|
ln
x
|
{\displaystyle \int {\frac {dx}{x\ln x}}=\ln \left|\ln x\right|}
∫
d
x
x
ln
x
ln
ln
x
=
ln
|
ln
|
ln
x
|
|
{\displaystyle \int {\frac {dx}{x\ln x\ln \ln x}}=\ln \left|\ln \left|\ln x\right|\right|}
, dst.
∫
d
x
x
ln
ln
x
=
li
(
ln
x
)
{\displaystyle \int {\frac {dx}{x\ln \ln x}}=\operatorname {li} (\ln x)}
∫
d
x
x
n
ln
x
=
ln
|
ln
x
|
+
∑
k
=
1
∞
(
−
1
)
k
(
n
−
1
)
k
(
ln
x
)
k
k
⋅
k
!
{\displaystyle \int {\frac {dx}{x^{n}\ln x}}=\ln \left|\ln x\right|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(n-1)^{k}(\ln x)^{k}}{k\cdot k!}}}
∫
d
x
x
(
ln
x
)
n
=
−
1
(
n
−
1
)
(
ln
x
)
n
−
1
{\displaystyle \int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}}
untuk
n
≠
−
1
{\displaystyle n\neq -1}
∫
ln
(
x
2
+
a
2
)
d
x
=
x
ln
(
x
2
+
a
2
)
−
2
x
+
2
a
tan
−
1
x
a
{\displaystyle \int \ln(x^{2}+a^{2})\,dx=x\ln(x^{2}+a^{2})-2x+2a\tan ^{-1}{\frac {x}{a}}}
∫
x
x
2
+
a
2
ln
(
x
2
+
a
2
)
d
x
=
1
4
ln
2
(
x
2
+
a
2
)
{\displaystyle \int {\frac {x}{x^{2}+a^{2}}}\ln(x^{2}+a^{2})\,dx={\frac {1}{4}}\ln ^{2}(x^{2}+a^{2})}
Integral yang melibatkan fungsi logaritmik dan trigonometri
(dengan asumsi
x
>
0
{\displaystyle x>0}
, dan konstanta integrasi tidak diperlihatkankan)
∫
sin
(
ln
x
)
d
x
=
x
2
(
sin
(
ln
x
)
−
cos
(
ln
x
)
)
{\displaystyle \int \sin(\ln x)\,dx={\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))}
∫
cos
(
ln
x
)
d
x
=
x
2
(
sin
(
ln
x
)
+
cos
(
ln
x
)
)
{\displaystyle \int \cos(\ln x)\,dx={\frac {x}{2}}(\sin(\ln x)+\cos(\ln x))}
Integral yang melibatkan fungsi logaritmik dan eksponensial
(dengan asumsi
x
>
0
{\displaystyle x>0}
, dan konstanta integrasi tidak diperlihatkankan)
∫
e
x
(
x
ln
x
−
x
−
1
x
)
d
x
=
e
x
(
x
ln
x
−
x
−
ln
x
)
{\displaystyle \int e^{x}\left(x\ln x-x-{\frac {1}{x}}\right)\,dx=e^{x}(x\ln x-x-\ln x)}
∫
1
e
x
(
1
x
−
ln
x
)
d
x
=
ln
x
e
x
{\displaystyle \int {\frac {1}{e^{x}}}\left({\frac {1}{x}}-\ln x\right)\,dx={\frac {\ln x}{e^{x}}}}
∫
e
x
(
1
ln
x
−
1
x
(
ln
x
)
2
)
d
x
=
e
x
ln
x
{\displaystyle \int e^{x}\left({\frac {1}{\ln x}}-{\frac {1}{x(\ln x)^{2}}}\right)\,dx={\frac {e^{x}}{\ln x}}}
Pustaka
(Inggris) Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. A few integrals are listed on page 69 dalam buku klasik ini.
Kata Kunci Pencarian:
- Daftar integral dari fungsi logaritmik
- Logaritma
- Tabel integral
- Integral tak tentu
- Daftar topik kalkulus
- Indeks artikel logaritma
- Daftar fungsi matematika
- Pendiferensialan logaritmik
- Fungsi invers trigonometri
- Daftar identitas logaritma
