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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) = f '(x)/f(x).
 The natural logarithm can be integrated using integration by parts:
$\int ln(x)dx = xln(x)  x + C$ .  The derivative of the natural logarithm eads to the Taylor series for ln(1 + x) around 0:
$ln(1+x) = x  \frac{x^{2}}{2} + \frac{x^{3}}{3}  ... for \left  x \right  \leq 1$ unless x = 1.
Terms

transcendental
of or relating to a number that is not the root of any polynomial that has positive degree and rational coefficients

irrational
of a real number, that cannot be written as the ratio of two integers
Full Text
The natural logarithm, generally written as ln(x), is the logarithm with the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.
The derivative of the natural logarithm is given by
This leads to the Taylor series for ln(1 + x) around 0:
is a picture of ln(1 + x) and some of its Taylor polynomials around 0. These approximations converge to the function only in the region −1 < x ≤ 1; outside of this region the higherdegree Taylor polynomials are worse approximations for the function.
Interactive Graph: Taylor Polynomial
The Taylor polynomials for only provide accurate approximations in the range −1 < x ≤ 1. Note that, for x > 1, the Taylor polynomials of higher degree are worse approximations.
Substituting x − 1 for x, we obtain an alternative form for ln(x) itself:
By using Euler transform, we reach the following equation, which is valid for any x with absolute value greater than 1:
The natural logarithm allows simple integration of functions of the form g(x) = f '(x)/f(x): an antiderivative of g(x) is given by ln(f(x)). This is the case because of the chain rule and the following fact:
In other words,
and
Here is an example in the case of g(x) = tan(x):
Letting f(x) = cos(x) and f'(x)= – sin(x):
where C is an arbitrary constant of integration.
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 The Substitution Rule
 approximation
 Appears in these related concepts: Area Expansion, Numerical Integration, and Roundoff Error
 chain rule
 Appears in these related concepts: Derivatives of Logarithmic Functions, Basic Integration Principles, and Directional Derivatives and the Gradient Vector
 converge
 Appears in these related concepts: Indeterminate Forms and L'Hôpital's Rule, Comparison Tests, and Convergence of Series with Positive Terms
 derivative
 Appears in these related concepts: Exponential and Logarithmic Functions, Antiderivatives, and Defining Financial Leverage
 differentiation
 Appears in these related concepts: Considering the Environment, Defining Monopoly, and Socioemotional Development in Adolescence
 e
 Appears in these related concepts: Derivatives of Exponential Functions, Natural Logarithms, and e
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Modal Mixture
 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 Graphs of Logarithmic Functions
 natural logarithm
 Appears in these related concepts: Logarithmic Functions, The Integral Test and Estimates of Sums, and Special Logarithms
 polynomial
 Appears in these related concepts: Polynomials: Introduction, Addition, and Subtraction, Partial Fractions, and Greatest Common Factor and Factoring by Grouping
 series
 Appears in these related concepts: Taylor Polynomials, Charging a Battery: EMFs in Series and Parallel, and Resisitors in Series
Sources
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Cite This Source
Source: Boundless. “The Natural Logarithmic Function: Differentiation and Integration.” Boundless Calculus. Boundless, 21 Jul. 2015. Retrieved 22 Jul. 2015 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/inversefunctionsandadvancedintegration3/inversefunctionsexponentiallogarithmicandtrigonometricfunctions13/thenaturallogarithmicfunctiondifferentiationandintegration812797/