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The Natural Logarithmic Function: Differentiation and Integration
Differentiation and integration of natural logarithms is based on the property
Learning Objective

Practice integrating and differentiating the natural logarithmic function
Key Points
 The natural logarithm allows simple integration of functions of the form
$g(x) = \frac{ f '(x)}{f(x)}$ .  The natural logarithm can be integrated using integration by parts:
$\int\ln(x)dx=x \ln(x)−x+C$ .  The derivative of the natural logarithm leads to the Taylor series for
$\ln(1 + x)$ around$0$ :$\ln(1+x) = x  \frac{x^{2}}{2} + \frac{x^{3}}{3}  \cdots$ for$\left  x \right  \leq 1$ (unless$x = 1$ ).
Terms

irrational
of a real number, that cannot be written as the ratio of two integers

transcendental
of or relating to a number that is not the root of any polynomial that has positive degree and rational coefficients
Full Text
The natural logarithm, generally written as
The derivative of the natural logarithm is given by:
This leads to the Taylor series for
for
Taylor Series Approximations for $\ln(1+x)$
The Taylor polynomials for
Substituting
for
By using Euler transform, we reach the following equation, which is valid for any
The natural logarithm allows simple integration of functions of the form
In other words:
and
Here is an example in the case of
Letting
where
The natural logarithm can be integrated using integration by parts:
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Key Term Reference
 antiderivative
 Appears in these related concepts: Numerical Integration, The Definite Integral, and The Fundamental Theorem of Calculus
 chain rule
 Appears in these related concepts: Directional Derivatives and the Gradient Vector, Related Rates, and The Substitution Rule
 derivative
 Appears in these related concepts: Integration By Parts, Separable Equations, and Overview of Derivatives
 differentiation
 Appears in these related concepts: Development of Nervous Tissue, Caulobacter Differentiation, and The Challenge of Competition
 e
 Appears in these related concepts: Derivatives of Exponential Functions, Derivatives of Logarithmic Functions, and The Natural Exponential Function: Differentiation and Integration
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Functions and Their Notation
 integration
 Appears in these related concepts: Basic Integration Principles, Area and Distances, and Volumes of Revolution
 logarithm
 Appears in these related concepts: Bases Other than e and their Applications, Logarithmic Functions, and Odds Ratios
 natural logarithm
 Appears in these related concepts: The Integral Test and Estimates of Sums, Natural Logarithms, and Special Logarithms
 series
 Appears in these related concepts: Charging a Battery: EMFs in Series and Parallel, Finding the General Term, and APA: Series and Lists
Sources
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Cite This Source
Source: Boundless. “The Natural Logarithmic Function: Differentiation and Integration.” Boundless Calculus. Boundless, 30 Mar. 2016. Retrieved 01 May. 2016 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/inversefunctionsandadvancedintegration3/inversefunctionsexponentiallogarithmicandtrigonometricfunctions13/thenaturallogarithmicfunctiondifferentiationandintegration812797/