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The Natural Logarithmic Function: Differentiation and Integration
Differentiation and integration of natural logarithms is based on the property $\frac{d}{dx}ln(x) = \frac{1}{x}$ .
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.
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 this related concepts: Numerical Integration, Area and Distances, and The Fundamental Theorem of Calculus
 approximation
 Appears in this related concepts: Numerical Integration, Roundoff Error, and Uniform Electric Field
 chain rule
 Appears in this related concepts: Directional Derivatives and the Gradient Vector, Related Rates, and The Substitution Rule
 converge
 Appears in this related concepts: Comparison Tests, Summing an Infinite Series, and Convergence of Series with Positive Terms
 derivative
 Appears in this related concepts: Logistic Equations and Population Grown, Defining Financial Leverage, and Difference Quotients
 differentiation
 Appears in this related concepts: The Natural Exponential Function: Differentiation and Integration, Caulobacter Differentiation, and Considering the Environment
 e
 Appears in this related concepts: Derivatives of Exponential Functions, Natural Logarithms, and Special Logarithms
 function
 Appears in this related concepts: Limit of a Function, Average Value of a Function, and Modal Mixture
 integration
 Appears in this related concepts: Growth Strategy, Basic Integration Principles, and Consumer Purchasing Behavior
 logarithm
 Appears in this related concepts: Derivatives of Logarithmic Functions, Bases Other than e and their Applications, and Logarithms of Powers
 natural logarithm
 Appears in this related concepts: The Integral Test and Estimates of Sums, Further Transcendental Functions, and Converting between Exponential and Logarithmic Equations
 polynomial
 Appears in this related concepts: Cylinders and Quadric Surfaces, Differentiation Rules, and Basics of Graphing Polynomial Functions
 series
 Appears in this related concepts: Taylor Polynomials, Resisitors in Series, and Finding the General Term
Sources
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Cite This Source
Source: Boundless. “The Natural Logarithmic Function: Differentiation and Integration.” Boundless Calculus. Boundless, 02 Jul. 2014. Retrieved 24 Mar. 2015 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/inversefunctionsandadvancedintegration3/inversefunctionsexponentiallogarithmicandtrigonometricfunctions13/thenaturallogarithmicfunctiondifferentiationandintegration812797/