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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

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 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 Indefinite Integrals and the Net Change Theorem
 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: Higher Derivatives, Separable Equations, and Logistic Equations and Population Grown
 differentiation
 Appears in this related concepts: Development of Nervous Tissue, The Many Goals of Persuasion, and Socioemotional Development in Adolescence
 e
 Appears in this related concepts: Derivatives of Exponential Functions, Derivatives of Logarithmic Functions, and Natural Logarithms
 function
 Appears in this related concepts: Solving Differential Equations, Four Ways to Represent a Function, and Average Value of a Function
 integration
 Appears in this related concepts: Growth Strategy, The Definite Integral, and Consumer Purchasing Behavior
 logarithm
 Appears in this related concepts: Bases Other than e and their Applications, pOH and Other p Scales, and Logarithmic Functions
 natural logarithm
 Appears in this related concepts: The Integral Test and Estimates of Sums, Converting between Exponential and Logarithmic Equations, and Special Logarithms
 polynomial
 Appears in this related concepts: Differentiation Rules, Partial Fractions, and Greatest Common Factor and Factoring by Grouping
 series
 Appears in this 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, 02 Jul. 2014. Retrieved 27 May. 2015 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/inversefunctionsandadvancedintegration3/inversefunctionsexponentiallogarithmicandtrigonometricfunctions13/thenaturallogarithmicfunctiondifferentiationandintegration812797/