Exploring Integrals Involving Logarithmic Function by Different
Methods
Siyang Wang
a
Shenghua Zizhu Academy, Shanghai, China
Keywords: Definite Integral, Logarithmic Function, L'Hôpital's Rule.
Abstract: This paper focuses on the theoretical development and applications of integral and logarithmic functions, with
the research background covering the historical evolution of mathematical analysis and modern
interdisciplinary demands. From the foundational work to modern scientific and engineering applications, the
integral serves as a core tool for quantifying continuous variables. The logarithmic function, due to its data
compression and computational simplification characteristics, has become a universal language across
multiple disciplines. The paper emphasizes addressing the systematization of integral methods for logarithmic
functions, involving the solution of complex integral expressions and the handling of improper integrals. The
research method combines classical mathematical tools with innovative techniques: through integration by
parts, linear variable substitution and the residue theorem. L'Hôpital's rule and partial fraction decomposition
are utilized to handle limits and integrals with logarithms in the denominator. The research indicates that the
combination of integral and logarithmic functions provides mathematical support for fields. The significance
of the paper lies in integrating theory and application, strengthening the universality of integral techniques,
and building a rigorous framework for modelling complex continuous systems and solving practical problems,
promoting the in-depth application of mathematical tools in scientific and technological innovation.
1 INTRODUCTION
The concept of integration originated in ancient
civilizations. Archimedes of ancient Greece found a
method to calculate the areas of curves and volumes,
approaching the exact solution through infinite
subdivision. Liu Hui's "circle-cutting method" and the
volume formula for spheres by Zu Chongzhi and his
son in China also contained the idea of integration.
Newton and Leibniz established calculus in the 17th
century, defining integration as the inverse operation
of differentiation and introducing the symbol ∫, laying
the foundation for modern integration theory. In the
19th century, Cauchy and Riemann refined the strict
mathematical definition of integration, making it a
core tool for analyzing continuous variables
(Atkinson & Han, 2012).
Integration is widely applied in science and
engineering. In physics, it is used to calculate the
work done which done by a variable force and the
distribution of electric fields. In engineering, it helps
determine the center of gravity and moment of inertia
a
https://orcid.org/0009-0009-2465-0730
of structures (Lu, 2025). In probability theory,
integration is employed to find the expected value of
continuous random variables, while in artificial
intelligence, it is utilized to optimize loss functions.
In finance, stochastic integration is used to simulate
stock price fluctuations, and in environmental
science, integration models are employed to predict
the spread of pollutants. From classical mechanics to
quantum computing, integration remains a bridge for
quantifying continuous changes and connecting
mathematics with reality, driving human
understanding of the complex world and
technological innovation (Stewart, 2015).
The logarithmic function was established by the
Scottish mathematician John Napier in 1614, aiming
to solve the complex multiplication and division
problems in astronomy (Dautov, 2021). His work
proposed the use of logarithms as an effective tool for
simplifying calculations, converting multiplication
and division into addition and subtraction, and
significantly enhancing efficiency with the aid of
logarithmic tables (Arfken et al, 2013). Subsequently,
104
Wang, S.
Exploring Integrals Involving Logarithmic Function by Different Methods.
DOI: 10.5220/0013814800004708
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy (IAMPA 2025), pages 104-108
ISBN: 978-989-758-774-0
Proceedings Copyright © 2025 by SCITEPRESS – Science and Technology Publications, Lda.

Henry Briggs developed common logarithms with
base 10 and compiled detailed logarithmic tables,
promoting their application in fields such as
navigation and engineering. In the 18th century, Euler
clarified the inverse relationship between logarithms
and exponents and introduced the natural logarithm
(with base e), laying the foundation for calculus and
scientific analysis.
The rest of this paper is organized as follows.
Section 2 describes the methodology and concepts,
Section 3 presents the integral results, and Section 4
concludes the paper.
2 METHODS
2.1 Integrating Logarithmic Functions
In modern applications, the logarithmic function
permeates multiple disciplines. The first is in
scientific measurement, decibels (sound intensity),
pH values (acidity and alkalinity), and the Richter
scale (earthquake energy) all use logarithmic scales to
compress a wide range of data. The second is in
economics and biology, compound interest models
and population growth are often described by
exponential functions (Malyavin, 2022). Taking the
logarithm of these functions linearizes the analysis.
The third is in information technology, algorithm
complexity (such as binary search 𝑂( log 𝑛)) and
data compression rely on logarithms to simplify
problem scales. The final is in engineering and
astronomy, signal attenuation and star brightness
calculations both require logarithmic conversions to
enhance data processing efficiency. The logarithmic
function has evolved from a practical computing tool
to a core language in scientific research, continuously
pushing the boundaries of human cognition and
technological development. (Smith, 1998).
Logarithmic function calculus is a significant part
of calculus, with its core revolving around the natural
logarithmic function ln𝑥 and its extended forms. In
terms of derivatives, the derivative of the natural
logarithmic function ln𝑥 is


, which is its most
notable property. For logarithmic functions with
general Ibases log

𝑥, they can be transformed into
natural logarithmic forms through the change-of-base
formula, and their derivatives are


. Regarding
integrals,
∫
ln𝜒𝑑𝑥 = ln𝜒 − 𝜒 + 𝐶 , while the
integral
∫


𝑑𝑥 = ln|𝑥| + 𝐶 reveals the intrinsic
connection between the natural logarithm and the
reciprocal function.
At the application level, logarithmic
differentiation is a key tool, suitable for simplifying
the differentiation process of power-exponential
functions (such as 𝑦=𝑓(𝑥)
()
or complex product
functions. By taking the logarithm of the function,
multiplication is transformed into addition, and then
the chain rule is used for differentiation, which can
efficiently handle complex expressions like 𝑦=


√



. Additionally, logarithmic functions are often
used in integration by substitution, for example, when
dealing with
∫


, setting 𝑢=ln𝑥 simplifies it to
ln
|
ln𝑥
|
+𝐶. Understanding these basic properties
lays a mathematical foundation for analyzing
exponential growth, probability models, and
engineering problems.
Integration by parts is a fundamental technique in
calculus derived from the product rule. It transforms
the integral of a product of functions into simpler
terms using the formula ∫𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑢 .
This method is particularly useful for integrals
involving products of algebraic, exponential,
logarithmic, or trigonometric functions. Strategic
selection of 𝑢 (to differentiate) and 𝑑𝑣 (to integrate)
is key, often guided by the LIATE rule (Logarithmic,
Inverse trigonometric, Algebraic, Trigonometric,
Exponential) (Han et al, 2024). By reducing complex
integrals to manageable forms, it enables solutions to
problems like ∫𝑥 𝑒ˣ 𝑑𝑥 or ∫𝑙𝑛(𝑥) 𝑑𝑥, making it
indispensable in advanced mathematics and applied
sciences.
The indefinite integral of the natural logarithmic
function ln𝑥 can be derived through integration by
parts
ln𝑥𝑑𝑥=𝑥ln𝑥−𝑥+𝐶
(
1
)
For derivation process, let 𝑢=𝑙𝑛𝑥, 𝑑𝑣 = 𝑑𝑥, then
𝑑𝑢 =


𝑑𝑥 , 𝑣=𝑥. Using the integration by parts
formula
∫
𝑢𝑑𝑣 = 𝑢𝑣 −
∫
𝑣𝑑𝑢 , so it is calculated that
∫
ln𝜒𝑑𝑥 = 𝜒 ln𝑥−
∫
𝑥∙


𝑑𝑥 = 𝑥 ln𝑥− 𝑥+ 𝐶.
For log

𝑥𝑑𝑥 ,by using the change-of-base
formula log

𝑥=


, the integral is transformed into
log

𝑥𝑑𝑥=
1
ln𝑎
(
𝑥ln𝑥−𝑥
)
+𝐶
(
2
)
The linear variable substitution is another way to do
it. For
∫
ln
(
𝑎𝑥+ 𝑏
)
𝑑𝑥 , let 𝑡=𝑎𝑥+𝑏, so 𝑑𝑡 =
𝑎𝑑𝑥, the integral is transformed into
1
𝑎
ln𝑡𝑑𝑡=
1
𝑎
(
𝑡ln𝑡−𝑡
)
+𝐶
=
𝑎𝑥+ 𝑏
𝑎
ln
(
𝑎𝑥+ 𝑏
)
−
𝑎𝑥+ 𝑏
𝑎
+𝐶.
Exploring Integrals Involving Logarithmic Function by Different Methods
105

2.2 Other Methods and Techniques
The L'Hôpital's rule is a useful method used to
calculate limits. Suppose one has two functions
𝑓
(
𝑥
)
and 𝑔
(
𝑥
)
which are differentiable in an open
interval containing a point 𝑎 (except possibly at 𝑎
itself), and lim
→
𝑓
(
𝑥
)
=lim
→
𝑔
(
𝑥
)
=0 𝑜𝑟 ±∞.
The author considers the limit lim
→

(

)

(

)
. By
Cauchy's Mean Value Theorem, for any 𝑥 in a
neighborhood of 𝑎, there exists a point 𝑐 between 𝑥
and 𝑎 such that

(

)

(

)

(

)

(

)
=
‘
(

)
’
(

)
. As 𝑥→ 𝑎,𝑐→𝑎. If
the limit lim
→
‘
(

)
’
(

)
exists or is ±∞, then lim
→

(

)

(

)
=
lim
→
‘
(

)
’
(

)
. This is the essence of L'Hôpital's rule,
providing a powerful tool to evaluate indeterminate
forms (Zhu, 2023).
For the Lagrange theorem, for a given planar arc
between two endpoints which there is at least one
point at which the tangent to the arc is parallel to the
line through its endpoints (Li, 2023), i.e.,
𝑓
(
𝑏
)
−𝑓
(
𝑎
)
𝑏−𝑎
=𝑓

(
𝑐
)(
3
)
The Residue Theorem in complex analysis
simplifies computing contour integrals of
meromorphic functions. For a function f(z) with
isolated singularities inside a closed contour 𝐶 , the
integral around 𝐶 equals 2𝜋𝑖 multiplied by the sum
of residues within 𝐶. A residue, extracted from the
Laurent series coefficient of (𝑧−𝑧

)

encapsulates
local behavior near singularities. This theorem
transforms intricate integrals into manageable residue
calculations, crucial for evaluating real integrals,
analyzing wave propagation, and solving differential
equations. Its power lies in linking global integration
to localized singularity data, making it indispensable
in physics, engineering, and mathematical research.
Cauchy's Integral Theorem states that for a
holomorphic function in a simply connected domain,
the integral over any closed contour is zero.
Established by Augustin-Louis Cauchy, it is central
to complex analysis. These principles enable efficient
evaluation of complex integrals, residue calculus, and
solutions to partial differential equations, forming the
cornerstone of analytic function theory.
3 APPLICATIONS
The first example is a basic integral
𝐼=𝑥ln𝑥𝑑𝑥
(
4
)
One can solve the integral by using the integration by
parts. Assume that 𝑢=ln𝑥 , so 𝑑𝑢 =


𝑑𝑥; assume
𝑑𝑣 = 𝑥𝑑𝑥 , so 𝑣=


𝑥

. By substituting into the
formula for integration by parts, the solution is that
𝑢𝑑𝑣=𝑢𝑣−𝑣𝑑𝑢
=
1
2
𝑥

ln𝑥 − 
1
2
𝑥

∙
1
𝑥
𝑑𝑥
=
1
2
𝑥

ln𝑥 − 
1
2
𝑥𝑑𝑥
=
1
2
𝑥

ln𝑥 −
1
4
𝑥

+𝐶
(
5
)
The second example is the multiplication of
polynomials and logarithms
𝐼=𝑥

ln𝑥𝑑𝑥
(
6
)
One can also solve the question by using the
integration by parts. Assume 𝑢=ln𝑥 , so 𝑑𝑢 =


𝑑𝑥; assume 𝑑𝑣 =𝑥

𝑑𝑥 , so 𝑣=



. By substituting
into the formula for integration by parts, the solution
is that
𝑥

ln𝑥𝑑𝑥 =
𝑥

3
ln𝑥 − 
𝑥

3
∙
1
𝑥
𝑑𝑥
=
𝑥

3
ln𝑥 −
1
3
𝑥

𝑑𝑥
=
𝑥

3
ln𝑥 −
1
3
+𝐶
(
7
)
The third example is integration with a logarithm
in the denominator, i.e.,
𝐼=
1
𝑥ln𝑥
𝑑𝑥
(
8
)
To solve the question, one can use the method of
variable substitution (Liu & Liu, 2024). Assume 𝑡=
ln𝑥, so 𝑑𝑡 =


𝑑𝑥 , and thus 𝑑𝑥=𝑥𝑑𝑡=𝑒

𝑑𝑡. The
original integration can turn into the final solution,
which is

1
𝑒

∙𝑡
∙𝑒

𝑑𝑡 = 
1
𝑡
𝑑𝑡 = ln
|
𝑡
|
+𝐶=ln|ln𝑥|+𝐶
The fourth example is a combination of
logarithms and fractions, i.e.,
𝐼=
ln
(
1+𝑥
)
𝑥

𝑑𝑥
(
9
)
Likewise, one can solve the formula by using the
integration by parts. Assume that 𝑢 =ln
(
1+𝑥
)
, so
𝑑𝑢 =


𝑑𝑥; assume that 𝑑𝑣 =



𝑑𝑥 , so 𝑣=−


.
Substituting the equation into the formula, it is thus
found that

ln
(
1+𝜒
)
𝜒

𝑑𝑥 = −
ln
(
1+𝜒
)
𝜒
+
1
𝜒
(
1+𝜒
)
𝑑𝑥
IAMPA 2025 - The International Conference on Innovations in Applied Mathematics, Physics, and Astronomy
106

Given that the fraction decomposition is


(

)
=


−


, the integral on the right-hand side is

1
𝜒
(
1+𝜒
)
𝑑𝑥=ln|𝜒|−ln|1+𝜒|+𝐶
Therefore, the final solution is that
𝐼=−
ln
(
1+𝜒
)
𝜒
+ln
𝜒
1+𝜒
+𝐶
(
10
)
The fifth integral is of absolute value
𝐼=ln
|
𝜒
|
𝑑𝑥
(
11
)
To solve the equation, one can use the case-by-
case discussion. First, when 𝑥 > 0, the integration is
the same with
∫
ln𝑥𝑑𝑥, the solution is 𝜒ln𝜒−𝜒+
𝐶. Second, when 𝑥<0, let 𝑡=−𝑥, so ln|𝑥| = ln|𝑡|
and 𝑑𝑥 = −𝑑𝑡. The integration turns into
−ln𝑡𝑑𝑡=−
(
𝑡ln𝑡−𝑡
)
+𝐶
=𝜒ln
(
−𝜒
)
−𝜒+𝐶
Regardless of whether 𝑥 is positive or negative, the
solution of integration can be presented as
𝐼=𝜒ln
|
𝜒
|
−𝜒+𝐶
(
12
)
The sixth definite Integrals and improper Integrals
is
𝐼=ln𝜒


𝑑𝑥
(
13
)
This is an improper integral (with 𝑥 = 0 as a
singular point). One can calculate the limit which is
lim
→
∫
ln𝑥


𝑑𝑥 = lim
→

𝑥ln𝑥−𝑥



. Substituting the
upper and lower limits, it is thus found that
lim
→

(
1∙0−1
)
−
(
𝑎ln𝑎−𝑎
)

=−1−lim
→
𝑎ln𝑎+
0. Calculating the limit by using L'Hôpital's rule,
which is
lim
→
𝑎ln𝑎=lim
→
ln𝑎
1
𝑎
=lim
→
1
𝑎
−
1
𝑎

=lim
→
(
−𝑎
)
=0
The final solution is therefore
ln𝜒


𝑑𝑥 = −1
(
14
)
The seventh is a higher-order logarithmic integral
𝐼=
(
ln𝜒
)

𝑑𝑥
(
15
)
It can be solved by applying integration by parts
twice. For the first integration by parts, one can
assume 𝑢=
(
ln𝜒
)

, so 𝑑𝑢 = 2ln𝜒 ∙


𝑑𝑥 ; assume
𝑑𝑣 = 𝑑𝑥 so 𝑣=𝜒. The integration turns into
that 𝜒
(
ln𝜒
)

−
∫
𝜒∙2𝜒∙


𝑑𝑥 = 𝜒
(
ln𝜒
)

−
2
∫
ln𝜒𝑑𝑥. Next, one can use the integration by parts
the second times to calculate
∫
ln𝑥𝑑𝑥, and the
solution is that
∫
ln𝑥𝑑𝑥 = 𝑥 ln𝑥− 𝑥+ 𝐶 .
Combining the solution together, it is that

(
ln𝜒
)

𝑑𝑥 = 𝜒
(
ln𝜒
)

−2
(
𝜒ln𝜒−𝜒
)
+𝐶
=𝜒
(
ln𝜒
)

−2𝜒ln𝜒+2𝜒+𝐶
(
16
)
4 CONCLUSIONS
This article systematically explores the historical
evolution, core methods, and interdisciplinary
applications of integrals and logarithmic functions.
The main content is divided into three parts: Firstly,
by reviewing the development of integral theory from
Archimedes' "method of exhaustion" to the
establishment of Newton-Leibniz calculus and then to
the rigorous definition by Cauchy-Riemann, it
clarifies the core position of integrals in quantifying
continuous variables. At the same time, the
development of logarithmic functions from Napier's
simplification of astronomical calculations to Euler's
introduction of natural logarithms demonstrates their
evolution. The combination of the two highlights the
practical significance of mathematical tools in fields.
The second part focuses on the mathematical methods
for integrating logarithmic functions, systematically
deriving the integral formulas for the natural
logarithm and the general logarithmic function.
Additionally, the application of L'Hôpital's rule and
partial fraction decomposition demonstrates the
synergy between limit analysis and algebraic
techniques. The third part verifies the effectiveness of
the method through seven typical examples, covering
definite integrals, improper integrals, and higher-
order integrals. These examples not only consolidate
the theoretical derivation but also reveal the
universality of the recursive solution rules and the
handling of absolute values. The research results
show that the combination of integrals and
logarithmic functions provides mathematical support
for the modeling of continuous systems.
Although this article systematically reviews the
methodological framework for integration and
logarithmic functions, there remains room for
expansion. Additionally, the paper focuses mainly on
classical integration techniques and pays less
attention to the auxiliary role of modern
computational tools, such as symbolic computation
software. The value of these tools in verification and
accelerating the solution process could be further
explored. Future research directions can be developed
from the following perspectives: Exploring the
combination of logarithmic functions and fractional
calculus to address the modeling needs of nonlocal
Exploring Integrals Involving Logarithmic Function by Different Methods
107

problems and extending integral methods to
stochastic differential equations or high-dimensional
optimization problems in machine learning to
enhance the adaptability of theoretical tools.
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