Watch
Watching this resources will notify you when proposed changes or new versions are created so you can keep track of improvements that have been made.
Favorite
Favoriting this resource allows you to save it in the “My Resources” tab of your account. There, you can easily access this resource later when you’re ready to customize it or assign it to your students.
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:
Assign just this concept or entire chapters to your class for free.
Key Term Reference
 antiderivative
 Appears in these related concepts: Numerical Integration, The Fundamental Theorem of Calculus, and Indefinite Integrals and the Net Change Theorem
 approximation
 Appears in these related concepts: Numerical Integration, Roundoff Error, and Uniform Electric Field
 chain rule
 Appears in these related concepts: Directional Derivatives and the Gradient Vector, Related Rates, and The Substitution Rule
 converge
 Appears in these related concepts: Comparison Tests, Summing an Infinite Series, and Convergence of Series with Positive Terms
 derivative
 Appears in these related concepts: Separable Equations, Antiderivatives, and Types of Financial Markets
 differentiation
 Appears in these related concepts: Caulobacter Differentiation, The Challenge of Competition, and Considering the Environment
 e
 Appears in these related concepts: Derivatives of Exponential Functions, e, and Special Logarithms
 function
 Appears in these related concepts: Inverse Functions, Four Ways to Represent a Function, and Limit of a Function
 integration
 Appears in these related concepts: Growth Strategy, Basic Integration Principles, and Area and Distances
 logarithm
 Appears in these related concepts: pOH and Other p Scales, Changing Logarithmic Bases, and Logarithms of Powers
 natural logarithm
 Appears in these related concepts: The Integral Test and Estimates of Sums, Natural Logarithms, and Converting between Exponential and Logarithmic Equations
 polynomial
 Appears in these related concepts: The Remainder Theorem and Synthetic Division, Finding Polynomials with Given Zeroes, and Polynomials: Introduction, Addition, and Subtraction
 series
 Appears in these related concepts: Charging a Battery: EMFs in Series and Parallel, Resisitors in Series, and Finding the General Term
Sources
Boundless vets and curates highquality, openly licensed content from around the Internet. This particular resource used the following sources:
Cite This Source
Source: Boundless. “The Natural Logarithmic Function: Differentiation and Integration.” Boundless Calculus. Boundless, 01 Jul. 2015. Retrieved 01 Jul. 2015 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/inversefunctionsandadvancedintegration3/inversefunctionsexponentiallogarithmicandtrigonometricfunctions13/thenaturallogarithmicfunctiondifferentiationandintegration812797/