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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 Fundamental Theorem of Calculus, and Indefinite Integrals and the Net Change Theorem
 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: Separable Equations, Overview of Derivatives, and Financial Leverage
 differentiation
 Appears in these related concepts: WBC Formation, Considering the Environment, and Socioemotional Development in Adolescence
 e
 Appears in these related concepts: Natural Logarithms, The Number e, and Common Bases of Logarithms
 function
 Appears in these related concepts: Solving Differential Equations, Visualizing Domain and Range, and The Vertical Line Test
 integration
 Appears in these related concepts: Area and Distances, The Definite Integral, and Volumes of Revolution
 logarithm
 Appears in these related concepts: Solving Problems with Logarithmic Graphs, Changing Logarithmic Bases, and Logarithms of Powers
 natural logarithm
 Appears in these related concepts: The Integral Test and Estimates of Sums, Further Transcendental Functions, and Converting between Exponential and Logarithmic Equations
 series
 Appears in these related concepts: Combination Circuits, Resisitors in Series, and The General Term of a Sequence
Sources
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Cite This Source
Source: Boundless. “The Natural Logarithmic Function: Differentiation and Integration.” Boundless Calculus. Boundless, 26 May. 2016. Retrieved 27 Jul. 2016 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/inversefunctionsandadvancedintegration3/inversefunctionsexponentiallogarithmicandtrigonometricfunctions13/thenaturallogarithmicfunctiondifferentiationandintegration812797/