Application of Feynman’s Integral Technique to Representative
Integrals and Real Life Scenarios
Zonglin Li
a
Anglo-Chinese School (Independent), Singapore, Singapore
Keywords: Definite Integral, Feynman’s Integral, Dirichlet Integral.
Abstract: Calculus is a branch of mathematics that studies continuous change and motion, focusing on two fundamental
concepts: differential calculus and integral calculus. This paper specifically focuses on integral calculus,
which targets the accumulation of quantities and area under curves. Integrals play important roles in many
fields, and this is because they possess key characteristics like precision, flexibility and universality. Initially,
the great scientists Newton and Leibniz introduced the Newton-Leibniz formula to compute integrals but as
time progress, it is insufficient to tackle complicated integrals. Therefore, a technique called Feynman’s
integral technique will be introduced in this paper. This technique was originated from the Leibniz integral
rule and involves differentiation under the integral sign. By applying this technique, complex integrals can be
simplified as the integral is being converted to a differential equation. In this paper, how Feynman’s integral
technique is applied will be demonstrated with detailed examples, which covers a range of different types of
integrals, including some classic examples. This paper contributes to extending the idea of integral calculation,
facilitates the efficient solution of integral calculations in practical problems and real world application of
Feynman’s integral technique.
1 INTRODUCTION
Integration, the process of finding the antiderivative
of a function, is a fundamental concept in calculus,
primarily divided into two types: indefinite and
definite integrals. The theory of integration has great
importance in mathematical analysis, in fact, in is the
one of the twin pillars on which analysis is built. For
example, integral is ideal for modelling dynamic
systems like fluid flow and heat diffusion as integrals
work with continuous functions unlike discrete sums.
The theory was originated by the great Newton and
Leibniz over three centuries ago, made rigorous by
Riemann in the middle of the nineteenth century, and
extended by Lebesgue at the beginning of the
twentieth century. The fundamental theorem of
calculus, the Newton-Leibniz formula is
𝑓
(
𝑥
)
𝑑𝑥=𝐹
(
𝑏
)
−𝐹
(
𝑎
)
, where f
(
𝑥
)
=𝐹
(
𝑥
)
.
From this equation, the following two formulas for
indefinite and definite integral respectively can be
derived:
𝑓
(
𝑥
)
𝑑𝑥=𝐹
(
𝑥
)
+𝑐 and
𝑓
(
𝑥
)
𝑑𝑥=
a
https://orcid.org/ 0009-0007-6873-5352
lim
‖
∆
‖
→
∑
𝑓
(
𝑐
)
∆𝑥
. Integration are commonly used
to compute areas between curves or under a function,
calculate volume of solids and determine the length
of arcs and curve. It also has application in various
fields, such as physics, economics and even
environmental science and medicine.
However, there are many integrals that are very
tricky to solve merely using Newton-Leibniz formula
due to their complexity. Originally derived from the
Leibniz integral rule, Feynman’s integral technique is
a method that tackles a group of such integrals swiftly
(Wang et al, 2020). This group includes oscillatory
integrals, logarithmic integrals, Frullani-type
integrals, moment generating integrals and
conditionally convergent integrals. This technique
transforms such challenging integrals into
manageable forms by leveraging parameterization
and differentiation. Due to its nature of simplifying
complex integrals, Feynman’s integral trick plays
crucial roles in many fields. For example, quantum
mechanics in physics, control theory in engineering,
analytic number theory in pure mathematics and
more. The fundamental of Feynman’s integral
Li, Z.
Application of Feynman’s Integral Technique to Representative Integrals and Real Life Scenarios.
DOI: 10.5220/0013815100004708
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 119-123
ISBN: 978-989-758-774-0
Proceedings Copyright © 2025 by SCITEPRESS – Science and Technology Publications, Lda.
119
technique is to hinge on transforming a difficult
integral into a parameter-dependent function, then
leveraging differentiation to simplify it. By
introducing an artificial parameter a, the integral
becomes a function 𝐼(𝑎). Differentiating 𝐼(𝑎) with
respect to a often simplifies the integrand, allowing
the original integral to be recovered via integration
and boundary conditions.
This paper is going to cover the following
components: what is Feynman’s integral technique,
how does Feynman’s integral technique work with
the explanation of its applications in different
questions, and the real life applications of Feynman’s
integral technique in different fields, in this case,
precision physics, perturbation theory in quantum
chemistry and electromagnetism.
2 FEYNMAN’S INTEGRATION
METHOD
Feynman’s integration method, also known as
“differentiation under the integral sign”, is introduced
by renowned physicist Richard Feynman to tackle
complex integration questions in an unconventional
way. The main idea of this integration method is to
introduce an auxiliary parameter into the integral,
making it a function of the new parameter (Liu et al,
2024). This will simplify the integration process by a
large extent as the original integral can often be found
by substituting a specific value of the new function
due to the introduction of the parameter and thus the
final result of the original integral can be computed
by differentiating the new integral with respect to the
parameter, and integrating the result, then substituting
in the value that will transform the new integral to the
original integral.
The general formula for Feynman’s integral
technique is
𝑑𝐼
𝑑𝑎
=
𝜕
𝜕𝑎
𝑓
(
𝑥,𝑎
)
𝑑𝑥
(
1
)
where 𝑑 and 𝑐 are fixed limits. Consider this classic
example that is significantly simplified by Feynman’s
integration method to solve. This involves the
application of Feynman’s trick in improper integral
(Nahin, 2015). The typical example is
Ι=
sin𝑥
𝑥
𝑑𝑥
(
2
)
To evaluate this integral conventionally,
=
𝐼𝑚
is first to be taken. Then the function can be
written as
I=Im
𝑒
𝑥
𝑑𝑥
(
3
)
Due to the singularity at 𝑥 =0 , the contour
integration approach should be used. Let the function
𝑓
(
𝑧
)
=
, and integrate it over a keyhole contour in
the complex plane. The contour consists of a small
semicircle of radius 𝜖 around the origin, which avoids
singularity at 𝑧=0, a large semicircle of radius 𝑅 in
the upper-half plane and two straight lines along the
real axis from 𝜖→𝑅 and from −𝑅→𝜖. Since this
function is analytic both on the contour and inside as
no poles are enclosed, Cauchy-residue theorem can
be applied and the integral of f
(
𝑧
)
over this enclosed
contour is
𝑓
(
𝑧
)
𝑑𝑧
=0
(
4
)
Now, the author shall consider the three
contributions. For the large semicircle,
|
𝑧
|
=𝑅, as
𝑅→∞, the integral over the large semicircle
vanishes as 𝑒
=𝑒
(
)
decays exponentially in
the upper half plane. For the small semicircle,
|
𝑧
|
=
𝜖, as 𝜖→0, the integral over the small semicircle
contributes −𝜋𝑖, which is half the residue at 𝑧=0.
Then for the straight lines, they combine to give
𝑑𝑥 +
𝑑𝑥
. By substituting x=-x into the
integral, it becomes
𝑑𝑥=2𝑖
𝑑𝑥
.
Taking the limit 𝑅→∞ and 𝜖→0, the integral will
become 2𝑖
𝑑𝑥
, and the equation
2𝑖
𝑑𝑥
−𝜋𝑖=0 can be formed. The result of
the original integral
𝑑𝑥
=
.
This process however, can be simplified by
merely introducing a new parameter, 𝑎 , to the
function such that it becomes
𝐼
(
𝑎
)
=
𝑠𝑖𝑛𝑥
𝑥
𝑒
𝑑𝑥
(
5
)
The next step is to differentiate I
(
𝑎
)
with respect to 𝑎:
𝑑𝐼
𝑑𝑎
=− sin𝑥
𝑒
𝑑𝑥
(
6
)
Then by integration by parts, the following can be
obtained:
= −
. To recover 𝐼
(
𝑎
)
, integrate
with respect to 𝑎 to get
𝐼
(
𝑎
)
=−
1
1+𝑎
𝑑𝑎= −tan
𝑎+𝑐
(
7
)
where 𝑐 is a constant.
In order to obtain 𝑐, take 𝑎→∞, this causes
tan
𝑎→
, and −𝑒
to strongly oppose the
IAMPA 2025 - The International Conference on Innovations in Applied Mathematics, Physics, and Astronomy
120
integrand, which results in 𝐼
(
𝑎
)
→0. Therefore,
−
+𝑐=0 and 𝑐=
, then 𝐼
(
𝑎
)
=
−tan
𝑎.
To convert 𝐼
(
𝑎
)
back to 𝐼, 𝑎=0 will be taken
and the function will look like
𝐼
(
0
)
=
𝜋
2
−tan
0=
𝜋
2
,
(
8
)
and 𝜋2
⁄
will be the final result of the integral
𝑑𝑥
(Nahin, 2015).
The author now considers another example
𝐼=
𝑥−1
ln𝑥
𝑑𝑥
,
(
9
)
which involves the use of Feynman’s trick in definite
integral. To use Feynman’s integral method to tackle
this question, a parameter, a, need to be first
introduced to the integral and the integral will then
become
𝐼
(
𝑎
)
=
𝑥
−1
ln𝑥
𝑑𝑥
(
10
)
Note that the original integral correspond to 𝐼
(
1
)
.
Then, differentiate 𝐼
(
𝑎
)
under the integral sign, with
respect to a to get
𝑑𝐼
𝑑𝑎
=
𝑑
𝑑𝑎
𝑥
−1
ln𝑥
𝑑𝑥=
𝜕
𝜕𝑎
𝑥
−1
ln𝑥
𝑑𝑥
(
11
)
The derivative will then get simplified as
𝑥
=
𝑥
ln𝑥 , and the ln𝑥 in the denominator will get
canceled out:
𝑑𝐼
𝑑𝑎
=𝑥
𝑑𝑥=
1
𝑎+1
(
12
)
To recover the function 𝐼
(
𝑎
)
,
needs to be
integrated
𝐼
(
𝑎
)
=
1
𝑎+1
𝑑𝑎=ln
(
𝑎+1
)
+𝐶
(
13
)
The evaluation of 𝐼
(
0
)
can be used to compute the
value of 𝐶 , 𝐼
(
0
)
=
𝑑𝑥=0
. Therefore, 𝐶=
0 and 𝐼
(
𝑎
)
=ln
(
𝑎+1
)
. Then substitute 𝑎=1 to
obtain the original integral and the final result would
be ln2 (Nahin, 2015; Zill, 2009).
Another example is the evaluation of the Gaussian
integral, which makes use of Feynman’s trick in
infinite integral. The Gaussian integral is essentially
I=
𝑒
𝑑𝑥 . To compute this integral a
parameter needs to be first introduced to the integral
by defining
I
(
𝑎
)
= 𝑒
𝑑𝑥
(
14
)
Differentiate 𝐼
(
𝑎
)
with respect to 𝑎 to get 𝐼
(
𝑎
)
=
−𝑥
𝑒
𝑑𝑥
. Then to recover 𝐼
(
𝑎
)
, integrate
𝐼′
(
𝑎
)
using integration by parts, which will give the
result, a first order differential equation that relates
𝐼′
(
𝑎
)
to 𝐼
(
𝑎
)
:
𝐼
(
𝑎
)
=−
1
2𝑎
𝐼
(
𝑎
)(
15
)
To solve this equation, it has to be rewritten in the
form of
=−
𝐼
(
𝑎
)
, and thus
(
)
=−
𝑑𝑎.
Then integrate both sides of the equation and the left
hand side of the equation will become ln
|
𝐼
(
𝑎
)|
+𝑐
and the right side of the equation becomes
−
ln|𝑎| + 𝑐. Combine the two together, it is found
that ln
|
𝐼
(
𝑎
)|
=−
ln|𝑎| + 𝐶, or alternatively,
|
𝐼
(
𝑎
)|
=𝑒
|
|
𝑒
.
(
16
)
Since 𝑒
|
|
=
|
𝑎
|
=
|
|
,
|
𝐼
(
𝑎
)|
=
|
|
,
𝐼
(
𝑎
)
=
√
, where 𝐶
=±𝑒
is a new constant. To
determine the value of 𝐶’, the initial condition shall
be used. Substitute 𝑎=1 into the original integral
and 𝐼
(
1
)
=
𝑒
𝑑𝑥
=
√
𝜋 will be obtained.
Then substitute 𝑎=1 into the solution, 𝐼
(
1
)
=
√
=
√
𝜋
, thus 𝐶
=
√
𝜋 , which means the final
computation of the integral 𝐼=
(Zill, 2009).
3 APPLICATIONS OF
FEYNMAN’S INTEGRAL
3.1 Application in Precision Physics
Modern particle physics is becoming extremely
precise and reliant on theoretical predictions for the
analysis and interpretation of experimental results,
which depends on the calculation of multi-loop
corrections to physical observables. However, with
the aid of Feynman’s integral trick, the evaluation of
multi-loop integral can be significantly simplified as
the trick combines denominators of propagators in
loop integrals into a single term (Wang et al, 2021).
In general, the purpose of using Feynman’s trick is to
transform the multi-propagator integral into a single
denominator integral, enable momentum shifts and
dimensional regularization and therefore simplifies
divergent integrals for renormalization.
Loop integral in quantum field theory refers to the
corrections to processes, for example, particle
interactions via virtual particles. These integrals
typically have the following three characteristics:
have multiple denominators, are divergent and have
great dependencies on momentum. Hence, the
parameterization character of Feynman’s integral
trick can combine the denominators and make the
Application of Feynman’s Integral Technique to Representative Integrals and Real Life Scenarios
121
calculation process much easier. Consider the
following loop example (Griffiths, 2018),
𝐼=
𝑑
𝑘
(
2𝜋
)
1
(
𝑘
−𝑚
)
(
𝑘+𝑞
)
−𝑚
(
17
)
Next, apply Feynman’s integral trick and combine
the denominators by setting the parameter 𝑥𝜖
0,1
(Zill, 2009), it is found that
(
)(
)
=
𝑑𝑥
(
)
∆
, where ∆=𝑚
−𝑥𝑞
(
1−𝑥
)
−𝑖𝜀.
Then shift the momentum by taking 𝑙=𝑘+𝑥𝑞,
𝐼=𝑑𝑥
𝑑
𝑙
(
2𝜋
)
1
(
𝑙
−∆
)
(
18
)
This shift takes out all the cross terms, leaving the
integral to be solely dependent on 𝑙
. To handle the
divergence property of the integral, wick rotation to
the Euclidean space (𝑘
→𝑖𝑘
) needs to be carried
out
𝑑
𝑘 →𝑖
𝑑
𝑘
. The denominator will become
𝑙
+∆. Last, carry out dimensional regularization and
evaluate the integral in 𝑑=4−𝜀 dimensions
𝑑
𝑙
(
2𝜋
)
1
(
𝑙
+∆
)
∝
Γ
2−
𝑑
2
(
4𝜋
)
∆
.
(
19
)
This isolates divergence as the poles in 𝜀, which
are canceled during renormalization. Feynman’s
integral trick plays a vital role in many aspects of
precision physics and here are the examples of two
areas where Feynman’s integral trick are used.
Firstly, Feynman’s integral trick is used in Higg
Boson production(LHC) as in processes like gluon-
gluon fusion, it is crucial for simplifying complex
loop integrals and enabling precise theoretical
predictions. The Higgs boson is predominantly
produced via gluon-gluon fusion (𝑔𝑔→𝐻) at the
LHC. This involves a loop of virtual particles (e.g.,
top quarks) due to the Higgs' strong coupling to heavy
particles (Anastasiou, 2014). The process of loop
integral complexity requires evaluating loop integrals
with propagators involving the top quark mass (mt)
and external momenta. Thus, at next-to-leading order
(NLO) or next-to-next-to-leading order (NNLO),
Feynman parametrization manages integrals with
additional propagators and phase-space constraints
(Schwartz, 2014). This reduces theoretical
uncertainties in Higgs cross-section predictions
to ∼1−2%∼1−2%, critical for LHC precision tests.
Secondly, Feynman’s integral trick is also extremely
useful in hadronic vacuum polarization (HVP)
(Aoyama et al, 2020). HVP is a quantum effect where
virtual quark-antiquark pairs polarize the vacuum,
modifying the photon propagator. This contributes to
key observables like the muon’s anomalous magnetic
moment (𝑔−2) and precision tests of the Standard
Model. The HVP tensor involves loop integrals with
quark propagators and photon propagators. These
integrals are divergent and require regularization and
renormalization. Therefore, with Feynman’s
parameterization, more accurate results can be
obtained with ease (Aoyama et al, 2020). Precise
HVP calculation has a significant impact on precise
physics as it is crucial for resolving the ∼3.7σ∼
3.7σ discrepancy between Standard Model
predictions and Fermilab/BNL experiments and
refining predictions for ZZ-boson masses, Higgs
couplings, and lepton universality.
3.2 Application in Perturbation Theory
Feynman’s integral trick that involves the
introduction of an auxiliary parameter can
significantly simplify the complex calculation in
perturbation theory and energy correction in quantum
chemistry. It can simplify the matrix elements in
perturbation theory as Calculating matrix elements of
the perturbing Hamiltonian H′ between unperturbed
states often involves challenging integrals. Therefore,
by introducing an auxiliary parameter into the integral
and differentiate the integral with respect to the
parameter, the integral can be reduced to a much
simpler form (Sakurai & Napolitano, 2021). For
example, introduce the parameter 𝜆 into an integral
and it becomes
𝜓
∗
Η
𝜓
𝑑𝜏=
𝑑
𝑑𝜆
𝜓
∗
𝑒
𝜓
𝑑𝜏|𝜆=0
(
20
)
The outcome of this is that it avoids direct
computation of complex integrals and enables
systematic computation of first-and-higher-order
energy correction. Feynman’s trick is also a useful
tool in variational perturbation theory, which is the
combination of variational principle and perturbation
theory and its purpose is to approximate the ground
state energy of a system with Hamiltonian Η
=Η
+
𝜆𝑉
, where Η
is solvable and 𝜆𝑉
is a perturbation. It
addresses two main issues: solving singular or high
dimensional integrals, and parameter optimization as
trial wavefunctions depend on variational parameters.
3.3 Application in Electromagnetism
Feynman's integral trick, or Feynman
parameterization, is a critical tool in quantum
electrodynamics (QED) for simplifying loop
integrals, such as those encountered in calculating the
electron self-energy (Liu et al, 2023). The electron
self-energy correction corresponds to a one-loop
Feynman diagram where an electron emits and
reabsorbs a virtual photon. This process introduces a
divergent integral, which Feynman's trick helps
IAMPA 2025 - The International Conference on Innovations in Applied Mathematics, Physics, and Astronomy
122
manage by restructuring the integrand. The divergent
integral for the self-energy correction takes the form:
(
𝑝
)
∝
𝑑
𝑘
(
2𝜋
)
𝛾
(
𝑝−𝑘+𝑚
)
𝛾
(
𝑘
−𝑚
+𝑖𝜀
)
(
𝑝−𝑘
)
+𝑖𝜀
where k is the loop momentum.
Feynman’s trick simplifies this by introducing
a Feynman parameter x to combine the
denominators:
1
𝐴𝐵
=
𝑑𝑥
𝑥𝐴 +
(
1−𝑥
)
𝐵
(
21
)
Applying this identity merges the propagators into a
single quadratic denominator:
𝑑𝑥
𝑑
𝑘
(
2𝜋
)
𝛾
(
𝑝−𝑘+𝑚
)
𝛾
𝑘
−2𝑥𝑝𝑘+𝑥
(
𝑝
−𝑚
)
+𝑖𝜀
.
Shifting the momentum 𝑘→𝑘+𝑥𝑝 linearizes the
denominator, simplifying the integral to a scalar form.
The divergence is then isolated into terms
like
𝑑
𝑘/𝑘
, which can be regularized using
dimensional regularization or cut off methods. This
restructuring reveals the ultraviolet (UV) divergence
as a pole in 𝜖=4−𝑑 (in dimensional
regularization), which is absorbed into
renormalization constants for the electron mass and
charge. Feynman’s method not only streamlines
calculations but also clarifies how divergences relate
to measurable quantities, enabling precise predictions
like the Lamb shift or the electron’s anomalous
magnetic moment. This approach exemplifies how
Feynman’s trick turns intractable integrals into
structured problems, bridging formal theory and
experimental reality in QED.
4 CONCLUSION
Feynman’s integral trick is vital in the field of
calculus as not only does it solve many complex
integral problems, but it is also revolutionary due to
the uniqueness and innovativeness of its concept,
which can possibly inspire future innovations. In this
paper, the examples used are all classic integrals that
have been pre-discussed. This paper explains the
fundamental concepts and related knowledge
regarding Feynman’s integral technique and how to
apply it in practical questions, with the aid of detailed
worked examples. Furthermore, this paper has also
exploited various different fields where the
application of Feynman’s integral technique is
required, such as physics and engineering, where
Feynman’s integral technique simplifies loop
calculation. Indeed, simply by introducing a
parameter, differentiate with respect to the parameter
and then restoring the integral, computation of
complex integrals becomes much more intuitive and
easier. The Feynman’s integral technique offers a
valuable approach for solving challenging integrals.
In conclusion, this paper elaborates on how to
approach integrals by Feynman’s integral technique
and how it can be applied in various situations.
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