Techniques and Theories in Evaluating Definite Integrals Involving
Trigonometric Functions
Zeming Tao
a
Sierra Canyon High School, Los Angeles, U.S.A.
Keywords: Definite Integrals, Trigonometric Functions, Real Analysis.
Abstract: Trigonometric integrals play a significant role in mathematical analysis, particularly in calculus, Fourier
analysis, and physics. Their evaluation requires systematic techniques due to the periodicity and oscillatory
nature of sine and cosine functions. This paper considers fundamental strategies for evaluating definite
trigonometric integrals, including symmetry properties, orthogonality relations, power reduction formulas,
and product-to-sum formulas. One of the primary challenges here is simplifying complex integrals of products
or powers of trigonometric functions. With reduction formulas and transformation identities, one can break
integrals down into manageable terms. In this article, these methods are demonstrated step-by-step with
examples, revealing how effectively they work on challenging integrals. The results show that the use of
orthogonality properties can often eliminate entire terms, greatly simplifying calculations. Power reduction
and product-to-sum identities also allow integration without messy algebraic manipulations. The results in
this work have broad applications in physics, engineering, and signal processing, where trigonometric
integrals are frequently used. This work provides a systematic way of integrating these integrals, reducing
computational time in a wide variety of scientific and mathematical applications.
1 INTRODUCTION
Trigonometric functions are the foundation of a lot of
mathematics, e.g., calculus, differential equations,
and Fourier analysis (Ely & Jones, 2023). The
periodic and oscillatory nature of trigonometric
functions makes them indispensable in physics,
engineering, and signal processing. One of the most
prominent fields where trigonometric functions
frequently appear is definite integrals, particularly
when waveforms are under analysis, boundary-value
problems are being solved, and Fourier coefficients
are being computed.
While they are important, trigonometric integrals
are not necessarily straightforward to calculate
directly, especially when they involve products or
powers of sine and cosine functions (Chen & Guo,
2024). Having more than one frequency or exponent
typically requires sophisticated techniques to
minimize computations. Thus, building systematic
methods for the calculation of such integrals is
essential in theoretical and applied mathematics.
a
https://orcid.org/0009-0008-1713-2874
*Corresponding author
Trigonometric integrals have been extensively
studied in mathematical analysis. Standard calculus
textbooks introduce basic trigonometric integration
techniques such as substitution and integration by
parts. However, more complex examples require
invocation of symmetry properties, orthogonality
relations, and algebraic manipulations, which have
been extensively employed in Fourier series and
mathematical physics (Du et al, 2023). The concept
of orthogonality, for instance, is a central element in
Fourier analysis insofar as sine and cosine function
integrals determine periodic function coefficients.
In engineering applications such as electrical
engineering, trigonometric integrals usually appear in
signal processing, particularly in Fourier transforms
and filter design (Zhang, 2023). Furthermore, in wave
physics and quantum mechanics, definite
trigonometric integrals appear in solving
Schrödinger's equation and analyzing wave
interference patterns. Since these integrals play
crucial roles in numerous disciplines, efficient
evaluation methods are vital for problem
simplification and reduction of complexity.
90
Tao, Z.
Techniques and Theories in Evaluating Definite Integrals Involving Trigonometric Functions.
DOI: 10.5220/0013814500004708
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 90-93
ISBN: 978-989-758-774-0
Proceedings Copyright © 2025 by SCITEPRESS – Science and Technology Publications, Lda.

This paper is structured as follows. Section 1
introduces elementary techniques for finding definite
trigonometric integrals, including symmetry and
orthogonality relations. Section 2 discusses power
reduction formulas, which reduce integrals of even
powers of sine and cosine functions. Section 3
addresses product-to-sum identities, which transform
products of sine and cosine into sums of simpler terms.
Section 4 provides applications of these methods,
demonstrating their use in solving challenging
integrals. Finally, Section 5 concludes with a
conclusion of findings and also with some potential
lines of future research.
2 FUNDAMENTAL TECHNIQUES
Trigonometric integrals with definite values are a
fundamental topic in mathematical calculus and
analysis. They frequently arise in applied physics,
engineering, and mathematics, particularly in signal
processing, Fourier analysis, and mechanics (Fan,
2015). The main challenge of the evaluation of these
integrals is that the sine and cosine functions are
periodic and oscillating. Therefore, there must be a
method of reduction and evaluation of them in a
systematic manner.
2.1 Symmetry and Orthogonality
Properties
One of the most powerful tools in computing definite
integrals of trigonometric functions is the use of
symmetry and orthogonality. The sine and cosine
functions exhibit even and odd symmetry properties:
𝑠𝑖𝑛 𝑥 is an odd function, meaning sin
(
−𝑥
)
=
−sin𝑥 .In contrast, 𝑐𝑜𝑠 𝑥 is an even function,
meaning cos
(
−𝑥
)
=cos𝑥 (Jeffrey, 1995).
These properties are important when dealing with
definite integrals on symmetric intervals. If is odd, its
definite integral over a symmetric interval is zero:

𝑓
(
𝑥
)
𝑑𝑥=0


(if f(x) is odd). For example,

𝑥

sin𝑥 𝑑𝑥


=0. Since the function 𝑥

sin𝑥 is
odd, the integral calculates immediately to zero
without further computation (Hedayatian & Faghih
Ahmadi, 2007).
Another important property is that sine and cosine
functions are orthogonal on symmetric intervals (Liu
& Liu, 2024). For integers 𝑚 and 𝑛:
sin
(
𝑚𝑥
)
sin
(
𝑛𝑥
)
𝑑𝑥=
𝜋 𝑖𝑓 𝑚=𝑛≠0
0 𝑖𝑓 𝑚≠𝑛


(
1
)
Similar result for cosine functions:
 cos
(
𝑚𝑥
)
cos
(
𝑛𝑥
)
𝑑𝑥=
𝜋 𝑖𝑓 𝑚=𝑛≠0
0 𝑖𝑓 𝑚≠𝑛


(
2
)
This orthogonality test is fundamental to Fourier
series and enables one to easily calculate many
definite integrals by observing when terms reduce to
zero.
2.2 Reduction Formulas
Integrals of sine and cosine functions with powers
require to be minimized using techniques in order to
reduce computation. The power reduction formulae
express higher powers in terms of lower ones so that
integration can be carried out step by step (Chen et al,
2019). The following equations are useful in doing so:
sin

𝑥=
1 − cos2𝑥
2
,cos

𝑥=
1 + cos2𝑥
2
(
3
)
Now one can break down integrals with even
powers. For example:
𝐼=sin

𝑥 𝑑𝑥


(
4
)
The author shall start by using sin

𝑥=(sin

𝑥)

and
substituting the identity sin

𝑥=(


)

=


(1 −
2cos2𝑥+ cos

2𝑥). Using the identity for 𝑐𝑜𝑠

𝑥, one
can simplify cos

2𝑥=


(Yan, 2019). Thus,
the integral becomes
𝐼=
1
4


1 − 2cos2𝑥 +
1 + cos4𝑥
2
𝑑𝑥
(
5
)
Splitting it into separate terms, then 𝐼=


(

1 𝑑𝑥−


2

cos2𝑥 𝑑𝑥+



(
1 + cos4𝑥
)
𝑑𝑥




. Because

cos2𝑥 𝑑𝑥 =0


and

(
1 + cos4𝑥
)
𝑑𝑥


=0, then it
is found that
𝐼=
1
4

𝜋+
𝜋
2

=
3𝜋
8
.
(
6
)
2.3 Product-to-Sum Identities
For integrals of products of sine and cosine
functions with different arguments, direct integration
is not generally feasible. Instead, the product-to-sum
identities transform them into a sum of simpler
trigonometric terms:
sin𝐴cos𝐵=
1
2
(sin
(
𝐴+𝐵
)
+sin
(
𝐴−𝐵
)
.
(
7
)
For example, people can consider:
𝐼= cos3𝑥cos5𝑥 𝑑𝑥


(
8
)
Using the identity cos𝐴cos𝐵=


(
cos
(
𝐴−𝐵
)
+
cos
(
𝐴+𝐵
))
, one can rewrite that cos3𝑥cos5𝑥 =


(
cos2𝑥+cos8𝑥
)
. Then, the integral becomes
Techniques and Theories in Evaluating Definite Integrals Involving Trigonometric Functions
91

𝐼=
1
2
(
cos2𝑥+cos8𝑥
)
𝑑𝑥.
(
9
)
Since both cos2𝑥 and cos8𝑥 integrate to 0 over

0,𝜋

, the result is 𝐼=0. This identity is extremely
useful in physics and signal processing, where
trigonometric product integrals occur frequently in
Fourier analysis and wave equations.
3 RESULTS AND APPLICATIONS
The techniques for integrating definite integrals of
trigonometric functions, such as symmetry,
orthogonality, power reduction formulas, and product
to sum identities, provide people with a powerful
collection of tools for simplifying complex integrals.
Not only do these techniques make calculations
easier, but they also have significant applications in
the sciences and engineering.
3.1 Examples of Fourier Analysis
In Fourier analysis, one typically need to deal with
integrals of products of sine and cosine functions.
Symmetry and orthogonality characteristics of these
functions play a significant role in the simplification
of these integrals. For instance, orthogonality of sine
and cosine functions over symmetric intervals is
crucial in the decomposition of periodic signals into
their frequency contents (Lu, 2025). For example:
𝐼=𝑥𝑐𝑜𝑠
(
3𝑥
)
sin
(
4𝑥
)
𝑑𝑥


(
10
)
The integrand is the product of 𝑥, which is an odd
function, and cos
(
3𝑥
)
sin
(
4𝑥
)
. To simplify it, one
first use a product-to-sum identity for sine and cosine:
𝑐𝑜𝑠𝐴 𝑠𝑖𝑛𝐵=


(
sin
(
𝐴+𝐵
)
−sin
(
𝐴−𝐵
))
. Thus,
by substituting it into this equation, it is found that
cos
(
3𝑥
)
sin
(
4𝑥
)
=
1
2
(
sin
(
7𝑥
)
−sin
(
𝑥
))
Thus, the integral becomes:
𝐼= 𝑥
1
2
(
sin
(
7𝑥
)
−sin
(
𝑥
))
𝑑𝑥


(
11
)
Now, people can separate the integral: 𝐼=



𝑥sin
(
7𝑥
)


−



𝑥sin
(
𝑥
)


𝑑𝑥 Here, 𝑥sin
(
7𝑥
)
is an even function since 𝑥 is odd and sin
(
7𝑥
)
is also
odd. The product of two odd functions is an even
function, but the integral is over a symmetric interval

−𝜋,𝜋

, so the integral of any odd function over a
symmetric interval is zero. Hence:

𝑥sin
(
7𝑥
)
𝑑𝑥=0


. Similarly, 𝑥𝑠𝑖𝑛
(
𝑥
)
is also an
odd function, thus

𝑥sin
(
𝑥
)


𝑑𝑥=0. Since both
integrals are zero, the expression evaluates to 0.
3.2 Examples of Reduction Formulas
Here are two more complex examples utilizing
reduction formulas.
Example 1: 𝐼=

𝑠𝑖𝑛

𝑥


𝑑𝑥. To evaluate 𝐼, one
begins by expressing 𝑠𝑖𝑛

𝑥 as(𝑠𝑖𝑛

𝑥)

. Using the
reduction identity for 𝑠𝑖𝑛

𝑥, it is found that
𝑠𝑖𝑛

𝑥=
1 − cos
(
2𝑥
)
2


(
12
)
The cubic term can be expanded according to
𝑠𝑖𝑛

𝑥=


(1 − 3cos
(
2𝑥
)
+ cos

(
2𝑥
)
− cos

(2𝑥)).
Next, one uses the identity for cos

(
2𝑥
)
,
substituting in to the expression: 𝑠𝑖𝑛

𝑥=(1−
3cos
(
2𝑥
)
+


+


cos(4𝑥) − cos

(2𝑥)). Separating
the integrals, then
8𝑠𝑖𝑛

𝑥


=1𝑑𝑥


− 3 cos
(
2𝑥
)
𝑑𝑥


+
3
2
1𝑑𝑥


+
3
2
 cos
(
4𝑥
)
𝑑𝑥


− cos

(
2𝑥
)
𝑑𝑥


(
13
)
Since the integrals of cos
(
2𝑥
)
andcos
(
4𝑥
)
over

0,𝜋

are zero, the expression reduces to:
𝑠𝑖𝑛

𝑥


=
1
8
𝜋 +
3
2
𝜋 −   cos

(
2𝑥
)
𝑑𝑥





(
14
)
Further reduction of the

cos

(
2𝑥
)
𝑑𝑥


term can be
handled similarly, but the essential idea is that the
process simplifies the integral significantly.
Example 2:

cos

𝑥𝑑𝑥


. Using the power
reduction formula for cos

𝑥 and substituting it into
the expression for cos

𝑥, it is found that cos

𝑥=

()



. Expanding the square, it is calculated
that cos

𝑥=


(
1 + 2cos
(
2𝑥
)
+ cos

(2𝑥)
)
.
Next, applying the reduction formula for
cos

(
2𝑥
)
, it is found that
cos

𝑥=
1
4
1 + 2cos
(
2𝑥
)
+
1 + cos
(
4𝑥
)
2

(
15
)
Simplifying and separating it, it is inferred that
4 cos

𝑥𝑑𝑥


=
3
2
𝑑𝑥


+ 2 cos
(
2𝑥
)
𝑑𝑥


+
1
2
 cos
(
4𝑥
)
𝑑𝑥


(
16
)
The integrals of cos
(
2𝑥
)
and cos
(
4𝑥
)
over

0,𝜋

are
zero so the expression becomes:
 cos

𝑥𝑑𝑥


=
1
4

3
2
𝜋 =
3𝜋
8
(
17
)
IAMPA 2025 - The International Conference on Innovations in Applied Mathematics, Physics, and Astronomy
92

3.3 Examples of Product-to-Sum
Identities
Example: 𝐼=

cos3𝑥 cos7𝑥 sin5𝑥 𝑑𝑥


. This
integral involves the product of three trigonometric
functions. To evaluate it, one first breaks it down
using the product-to-sum identities (Shu, 2019).
Using the identity: cos𝐴cos𝐵=


(
cos(𝐴 − 𝐵) +
cos(𝐴 + 𝐵)
)
, it is inferred that for cos3𝑥 cos7𝑥,
cos3𝑥 cos7𝑥 =


(
cos4𝑥+ cos10𝑥
)
. Substituting
it into the integral:
𝐼=
1
2
(
cos4𝑥+ cos10𝑥
)
sin5𝑥𝑑𝑥


=
1
2
 cos4𝑥sin5𝑥𝑑𝑥


+
1
2
 cos10𝑥 sin5𝑥 𝑑𝑥


(
18
)
Using the identity:cos𝐴sin𝐵=


(
sin
(
𝐴+𝐵
)
−
sin(𝐴 − 𝐵
)
), then for each integral separately, it is
found that

cos4𝑥 sin5𝑥 𝑑𝑥


=





(
sin9𝑥−
sin𝑥
)
𝑑𝑥 and

cos10𝑥 sin5𝑥 𝑑𝑥


=



(sin15𝑥− sin5𝑥)𝑑𝑥


. Since the integral of any
sine function over a symmetric interval like

0,𝜋

is
zero, i.e.,

cos4𝑥 sin5𝑥𝑑𝑥


=0 as well as

cos10𝑥 sin5𝑥 𝑑𝑥


=0. Given that both terms are
zero, the original integral evaluates to:
𝐼=
1
2
(
0−0
)
=0.
(
19
)
4 CONCLUSIONS
This paper has explained major techniques for the
evaluation of definite integrals involving
trigonometric functions. By applying symmetry and
orthogonality principles, one showed how certain
integrals can be simplified or evaluated directly as
zero. Power reduction formulas were also useful in
simplifying higher powers of sine and cosine to lower
terms, facilitating easier integration. The application
of product-to-sum identities also allowed the
transformation of complex trigonometric products
into manageable expressions.
The results suggest the efficiency of these
methods to reduce trigonometric integrals, which is
practically useful in applications such as Fourier
analysis, signal processing, and physics. With the
help of these methods, it is possible to reduce
computational effort significantly in finding integral-
based problems.
Future research could extend to using these
methods on higher integrals of hyperbolic functions,
exponential functions, or multi-variable
trigonometric functions. Another option would be to
merge symbolic computation software, such as
Mathematica or MATLAB, with more efficient and
automated methods of calculating definite
trigonometric integrals. This methodology opens
doors for future investigation and application to more
advanced mathematical and engineering problems.
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