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integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s (
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
functions) of
logarithmic function In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
s. For a complete list of integral functions, see
list of integrals Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not ...
. ''Note:'' ''x'' > 0 is assumed throughout this article, and the
constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connected ...
is omitted for simplicity.


Integrals involving only logarithmic functions

: \int\log_a x\,dx = x\log_a x - \frac = \frac : \int\ln(ax)\,dx = x\ln(ax) - x : \int\ln (ax + b)\,dx = \frac : \int (\ln x)^2\,dx = x(\ln x)^2 - 2x\ln x + 2x : \int (\ln x)^n\,dx = x\sum^_(-1)^ \frac(\ln x)^k : \int \frac = \ln, \ln x, + \ln x + \sum^\infty_\frac : \int \frac = \operatorname(x), the
logarithmic integral In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...
. : \int \frac = -\frac + \frac\int\frac \qquad\mboxn\neq 1\mbox : \int \ln f(x)\,dx = x\ln f(x) - \int x\frac\,dx \qquad\mbox f(x) > 0\mbox


Integrals involving logarithmic and power functions

: \int x^m\ln x\,dx = x^\left(\frac-\frac\right) \qquad\mboxm\neq -1\mbox : \int x^m (\ln x)^n\,dx = \frac - \frac\int x^m (\ln x)^ dx \qquad\mboxm\neq -1\mbox : \int \frac = \frac \qquad\mboxn\neq -1\mbox : \int \frac = -\frac-\frac \qquad\mboxm\neq 1\mbox : \int \frac = -\frac + \frac\int\frac \qquad\mboxm\neq 1\mbox : \int \frac = -\frac + \frac\int\frac \qquad\mboxn\neq 1\mbox : \int \frac = \ln \left, \ln x\ : \int \frac = \ln \left, \ln \left, \ln x\ \, etc. : \int \frac = \operatorname(\ln x) : \int \frac = \ln \left, \ln x\ + \sum^\infty_ (-1)^k\frac : \int \frac = -\frac \qquad\mboxn\neq 1\mbox : \int \ln(x^2+a^2)\,dx = x\ln(x^2+a^2)-2x+2a\tan^ \frac : \int \frac\ln(x^2+a^2)\,dx = \frac \ln^2(x^2+a^2)


Integrals involving logarithmic and trigonometric functions

: \int \sin (\ln x)\,dx = \frac(\sin (\ln x) - \cos (\ln x)) : \int \cos (\ln x)\,dx = \frac(\sin (\ln x) + \cos (\ln x))


Integrals involving logarithmic and exponential functions

: \int e^x \left(x \ln x - x - \frac\right)\,dx = e^x (x \ln x - x - \ln x) : \int \frac \left( \frac-\ln x \right)\,dx = \frac : \int e^x \left( \frac- \frac \right)\,dx = \frac


''n'' consecutive integrations

For n consecutive integrations, the formula : \int\ln x\,dx = x(\ln x - 1) +C_ generalizes to : \int\dotsi\int\ln x\,dx\dotsm dx = \frac\left(\ln\,x-\sum_^\frac\right)+ \sum_^ C_ \frac


See also

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References

*
Milton Abramowitz Milton Abramowitz (19 February 1915 in Brooklyn, New York – 5 July 1958) was a Jewish American mathematician at the National Bureau of Standards who, with Irene Stegun, edited a classic book of mathematical tables called '' Handbook of Mathemati ...
and Irene A. Stegun, ''
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables ''Abramowitz and Stegun'' (''AS'') is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the ''National Institute of Standards and Te ...
'', 1964. A few integrals are listed o
page 69
{{Lists of integrals
Logarithmic functions In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...