Derivation Of Formulas For Evaluating Integrals Of
Powers And Products Of Hyperbolic Sine And
Hyperbolic Cosine
Renson Aguilar Robles
Batangas State University
Antipolo, San Pascual, Batangas, Philippines 4204
Abstract: The study aimed to develop formulas in evaluating integrals of powers and products of hyperbolic sine and hyperbolic cosine.
Integrals of the formsinh max cosh naxdx , where m or n Z;sinh maxdx and cosh naxdx , where Z n
m, and for all a except
zero were considered. Through basic knowledge on algebraic methods, differentiation and integration procedures, the study presented the generalized formulas in finite series forms in evaluating integrals of some hyperbolic functions. This study, as a pure research, is descriptive and expository in nature. This provided comprehensive description of the concepts in Mathematics, particularly on the context of hyperbolic functions, which may simplify formulas in evaluating hyperbolic integrals. The generalized formulas of hyperbolic integrals of various forms stated above were developed to shorten up the solutions in evaluating such integrals.
Keywords: Hyperbolic Cosine, Hyperbolic Sine, Powers of Hyperbolic Cosine, Powers of Hyperbolic Sine, Product of Hyperbolic Sine and Cosine.
1.
Introduction
One of the integral parts in the study of Calculus is hyperbolic functions. These functions are associated from the comparison of the area under a semicircular
region, 2
1 x
y with the area of the region under a
hyperbola 2
1 x
y [1]. The six hyperbolic functions
express in terms of x are sinhx, coshx, tanhx, cothx, sechx and cschx. These functions are now studied in a more advanced setting in most universities and colleges, and have found applications in the study of electricity [2] and catenary cable. Its properties and characteristics, derivatives and antiderivatives were studied and researches were conducted to exemplify its properties to apply it in solving problems in the fields of engineering and sciences. The study of [3] and [4] presented some applications of hyperbolic functions, particularly in the study of differential equations while the summation of series of hyperbolic functions was presented in [5]. Integrals of hyperbolic functions are also essential parts in the study of Calculus. Hyperbolic identities are used to evaluate integrals of the formsinh maxcosh naxdx , where m
or n Z; sinh maxdx and cosh naxdx , where m,nZ and
for all a except zero. The identity used depends on the
power of each function. If m or n is odd, the identity
1 sinh
cosh 2x 2x is applied. If m is even used the identity x
x cosh 2
2 1 2 1
cosh 2 and when n is even the identity
x
x cosh 2
2 1 2 1
sinh 2 is used. These identities are used to
reduce the powers of each function and the integrand is transformed into an expression where appropriate integration formulas can be easily applied. However, this method of finding the integrals of powers and products of hyperbolic sine and hyperbolic cosine becomes very tedious and time consuming for larger values of m and n. This paper aimed to develop formulas in finite series forms for evaluating
integrals of the form sinh maxcosh naxdx , where m or
n Z; sinh maxdx and cosh naxdx , where
Z n
m, and for
all a except zero. The derived formulas will help
students to avoid lengthy solutions in evaluating integrals of these hyperbolic functions. Also, this study will enrich the students’ knowledge and intensify learning on some concepts where hyperbolic function is an integral part.
2.
Derivation of Formulas
The main objective of the study was to develop formulas in finite series form for evaluating integrals of powers and products of hyperbolic sine and hyperbolic cosine. Rules on algebra, formulas on differentiation and techniques of integration were utilized to achieve this purpose.
2.1. sinh m ax cosh n axdx , where mis an odd positive and
a
n, except 0
When m is odd, factor out sinh ax and change the remaining
hyperbolic sine into hyperbolic cosine using the identity
1 cosh
sinh 2ax 2ax . Thus,
axdx ax ax
axdx ax ax
axdx ax ax axdx
ax
n m
n m
n m
n m
sinh cosh 1 cosh
sinh cosh sinh
sinh cosh sinh
cosh sinh
2 1 2
2 1 2
1
(1)
Let
2 1
m
k and kan even number, then
ax axdx ax nax axdx
k n
m
sinh cosh 1 cosh cosh
sinh 2 (2)
Let ucosh ax duasinh axdx
du axdx a
sinh 1
u u du
a axdx
ax n
k n
m
1 1 cosh
sinh 2
Expanding
u2 1
kusing Binomial Theorem, the right side of (3) becomes
u u du
k k u k u k u k u a axdx
ax n k k k k n
m 1 1 ... 3 2 1 1 cosh
sinh 2 2 1 2 2 2 3 2 (4)
Distributing un for each term inside the bracket of equation 4 and integrating,
C u n u n k k u n k k u n k k u n k k u n k a du u u k k u k u k u k u a axdx ax n n n k n k n k n k n n n k n k n k n k n m 1 1 2 1 3 2 1 2 2 1 1 2 1 2 2 3 2 2 2 1 2 2 1 1 1 2 1 1 ... 1 3 2 1 3 1 2 2 1 2 1 1 2 1 1 1 2 1 1 1 ... 3 2 1 1 cosh sinh (5)
Simplifying further, equation (5) becomes
C u n u n k k u n k k u n k k u n k k u n k a axdx ax n n n k n k n k n k n m 1 3 5 2 3 2 1 2 1 2 1 1 3 1 1 ... 5 2 1 3 3 2 1 2 1 2 1 1 1 2 1 1 cosh sinh (6) Since 2 1 m
k , then equation (6) becomes
C u n u n m m u n m m u n m m u n m m u n m a axdx ax n n n m n m n m n m n m 1 3 5 2 1 2 3 2 1 2 1 2 1 2 1 2 1 2 1 1 3 1 1 2 1 2 1 ... 5 2 1 2 1 3 2 1 3 2 1 2 1 2 2 1 1 2 1 2 1 1 2 1 1 2 1 2 1 1 cosh sinh (7)
Simplifying further, equation (7) will become
2 4 6
6 1 3 2 1 4 1 2 2 1 2 1 1 2 1 1 1 cosh
sinh m n m n m n m n um n
n m m u n m m u n m m u n m a axdx ax u C n u n m m n n
3 1
1 1 3 1 2 3 2 1
... (8)
In general, substituting ucosh ax in equation (8),
ax n m m ax n m m ax n m a axdx
ax n m n m n m n
m 2 4
cosh 4 1 2 2 1 cosh 2 1 1 2 1 cosh 1 1 cosh
sinh
C ax n ax n m m ax n m m n n n m
6 3 1
cosh 1 1 cosh 3 1 2 3 2 1 ... cosh 6 1 3 2 1 (9) The right side of equation (9) can now be written in finite series form as
ax C
p n m p k a axdx ax k p p n m p n m 0 2 cosh 2 1 1 1 cosh
sinh (10)
where
2 1
m
k and p 0,1,2,3,4,...,k ;
! ! ! p p k k p k
When n is odd positive integer, factor out cosh ax and then change the remaining hyperbolic cosine into hyperbolic sine using
the identity cosh 2ax sinh 2ax 1
axdx ax
ax axdx
ax n n m
m
cosh sinh
cosh cosh
sinh 1
ax m ax axdx
n
cosh sinh
cosh 2 1
2 (11)
Let
2 1
n
k and kan even number
ax axdx ax m ax axdxk n
m
cosh sinh
1 sinh cosh
sinh 2 (12)
Let usinh ax duacosh axdx
du axdx a
cosh 1
u u du
a axdx
ax m
k n
m
1 1 cosh
sinh 2 (13)
Expanding
k
u2 1 using Binomial Theorem, the right hand side of eq. (13) will become
du u u k
k u
k u
k u
k u a axdx
ax n k k k k m
m
1 1 ... 3
2 1
1 cosh
sinh 2 2 1 2 2 2 3 2 (14)
Distributing um for each term inside the bracket in eq. (14) and integrating,
du u u k
k u
k u
k u
k u
a axdx
ax n k n k m k m k m m m
m 2 2 1 2 2 2 3 2
1 ... 3
2 1
1 cosh
sinh
C u m u m k
k u
m k k
u m k k u
m k k u
m k a
m m
m k
m k m
k m
k
1 1
2 1
3 2
1 2 2 1
1 2 1
2
1 1
1 2
1
1 ... 1
3 2
1
3
1 2 2
1
2 1
1 2
1
1 1
2 1 1
(15)
Simplifying further eq. (15) will become
3 2 1
2 1
2
3 2
1 2 1
2 1 1 1
2 1 1 cosh
sinh
m n k m k m k m
u m k k u
m k k u
m k a axdx ax
u C
m u m k
k u
m k
k k m m m
2 5 3 1
1 1
3 1
1 ... 5
2 1
3 (16)
Since
2 1
n
k , then it follows that eq. (16) becomes
3 2
1 2 1
2 1 2 1
2 1 2
3 2
1 2
1
2 2 1
1 2
1 2
1
1 2 1
1 2
1 2
1 1
cosh sinh
m
n m
n m
n n
m
u
m n n
u
m n n
u
m n a
axdx
u C
m u m n
n
u
m n n
m m
m n
1 3
5 2
1 2
1 1 3
1 1 2
1 2 1 ...
5 2
1 2
1 3
2 1
(17)
Simplifying further, eq. (17) will become
2 4
4 1 2
2 1 2
1 1
2 1 1
1 cosh
sinh m n m n m n um n n
m n u
n m n u n m a axdx ax
u C
m u m n n
u n m n
m m
n
m
6 3 1
1 1 3
1 2
3 2
1 ... 6
1 3
2 1
(18)
C ax m
ax m
n n
ax n
m n
ax n
m n ax n
m n ax n
m a axdx ax
m m
n m
n m n
m n
m n
m
1 3
6
4 2
sinh 1 1 sinh
3 1 2
3 2
1 ... sinh
6 1 3
2 1
sinh 4 1 2
2 1 sinh
2 1 1
2 1 sinh
1 1 cosh
sinh
(19)
The right side of eq. (19) can now be expressed in finite series form as
C ax p
n m p k
a axdx ax
k
p
p n m n
m
0
2
sinh 2 1 1
cosh
sinh (20)
where
2 1
n
k and p 0,1,2,3,4,..., k ;
! !
!
p p k
k
p k
Another modification of (10) and (20) is that when the integrand is hyperbolic sine or hyperbolic cosine only with odd powers.
2.3. sinh m axdx , where m is odd positive integer and a not equal to 0
ax C
p m p k
a axdx
k
p
p m p
m
0
2
cosh 2 1 1
1
sinh ; (21)
where
2 1
m
k and p0,1,2,3,4,...,k;
! ! !
p p k
k
p k
2.4. cosh n axdx , where n is odd positive integer and a not equal to 0
ax C
p n p k
a axdx
k
p
p n
n
0
2
sinh 2 1 1
cosh (22)
where
2 1
n
k and p0,1,2,3,4,...,k ;
! ! !
p p k
k
p k
Consider the following derivation of formulas for powers of hyperbolic sine and hyperbolic cosine, i.e. when m or n is even. Here, the exponential identities
2 sinh
a x a x
e e ax
and
2 cosh
a x a x
e e ax
shall be used. 2.5. sinh m axdx , where m is an even positive integer and a not equal to 0
Since
2 sinh
a x a x
e e ax
, then
dx e e axdx
m ax ax m
2
sinh (23)
e e dx
m ax ax m
2 1
1 2 2 2 2 2 2
2 ... 2
... 2
1 2
1
sinh
ax axm
m ax m ax ax
m ax ax
m ax axm
m m
e e m
m e
e m m e
e m e e m e
axdx
dx e e
e m
m ax axm axm
1
1
dx e e
m m e
m m m
m
e m e
m
eaxm ax m ax m ax m ax m axm
m
2 4 4 2
1 2
...
2 ... 2
1 2
1
(24)
Regrouping term by term (such as the first and the last, the second and the second to the last so on and so forth) and since
1 1 m
m m
;
2 2 m
m m
;
3 3 m
m m
dx m m e
e m e
e m e
e
axdx axm axm axm axm axm axm
m m
2 ... 2
1 2
1
sinh 2 2 4 4
dx m m e
e m e
e m e
eaxm axm axm axm axm axm
m
2 ... 2
1 2
1 2
1 2 2 4 4
1
(25)
Distributing
2 1
to each term on the integrand (25) becomes
dx m m e
e m e
e m e
e axdx
m ax m ax m
ax m ax axm
axm m
m
2 2 1 ... 2
2 2
1 2
2 1 sinh
4 4
2 2
1
(26)
Since e e ax
ax ax
cosh 2
, equation (26) can be written as
2 2 1 2 cosh 1 2 ... 4 cosh
2 2 cosh
1 cosh
2 1 sinh
1
m m
ax m
m
m ax m m
ax m amx axdx
m
m (27)
Integrating (27) leads to
C x m m
ax a
m m
m ax m
a m m
ax m
a m amx am
axdx
m m
2 2 1 2 sinh 2
1
1 2 ...
4 sinh
4 1
2 2 sinh
2 1
1 sinh
1
2 1 sinh
1
(28)
In general, (28) can be simplified as
m x C
m
ax m
m
m ax m
m m ax m
m amx m
a axdx
m m
m
2 2 1 2 sinh 2 1
1 2 ... 4 sinh
4 1
2 2 sinh
2 1
1 sinh
1
2 1 1 sinh
1
(29)
x C
k m x
p m a p m p m
a axdx
m k k
p p m
m
2 1 1 2
sinh 2 1 1
2 1 1 sinh
1
0 1
(30)
where
2
m
k and p0,1,2,3,4,...,k ;
! ! !
p p m
m
p m
2.6. cosh n axdx , where n is an even positive integer and a not equal to 0
Following the same procedure employed in section 2.5, cosh n axdx can be derived as follows.
x C
k n x
p n a p n p n
a axdx
n k
p n n
2 1 2 sinh 2 1 2
1 1 cosh
1
0 1
(31)
where
2
n
k and p 0,1,2,..., k ;
! !
!
p p n
n
p n
Example 1. Evaluate sinh 52xcosh 22xdx
By applying formula (10), we have m 5; n2; a2 and 2 2
1 5
k ; p0,1,2
C x x
x
C x x
x
C x
x x
xdx x
2 cosh 6 1 2 cosh 5 1 2 cosh 14
1
2 cosh 3 1 2 cosh 5 1 2 2 cosh 7 1 2 1
2 cosh
2 2 2 5
1 2 2 1
2 cosh
1 2 2 5
1 1 2 1 2 cosh
0 2 2 5
1 0 2 1 2 1 2 cosh 2 sinh
3 5
7
3 5
7
2 2 2 5 2
1 2 2 5 1
0 2 2 5 0
Example 2. Evaluatesinh 43xcosh 53xdx
By applying formula (20), we have m4 ; n5; a3and 2 2
1 5
k ; p0,1,2
x xdx x sinh 3x
2 5 4
1
1 2 3 h sin 0 5 4
1 0 2
3 1 3 cosh 3
sinh 4 5 9 7 x C
sinh 3
4 5 4
1 2
2 5
C x x
x
C x x
x
C x x
x xdx
x
3 sinh 15
1 3 sinh 21
2 3 sinh 27
1
3 sinh 5 1 3 sinh 7 2 3 sinh 9 1 3 1
3 sinh 5 1 2 2 3 sinh 7 1 1 2 3 h sin 9 1 3 1 3 cosh 3 sinh
5 7
9
5 7
9
5 7
9 5
4
Example 3. Evaluate cosh 43xdx
By applying formula (31), we have a3; n4; k2
C x x x
C x x
x
C x x
x xdx
8 3 6 sinh 12
1 12 sinh 96
1
6 16
1 6 sinh 2 12 sinh 4 1 8 1 3 1
2 4 2 1 2 4 3 sinh 1 4
1 1
4 0 4 3 sinh 0 4
1 0 4 2 1 3 1 3 cosh
4 1
4 4
3. Conclusion
The developed formulas presented on this paper showed an easy way of evaluating integrals of powers and products of hyperbolic sine and cosine functions. By using these formulas, tedious repetition of integration procedures and expansion of identities using the traditional methods are eliminated. Integrals of such functions can easily be obtained directly since the formulas are expressed in finite series forms. Also, the derived formulas will be very useful to mathematics and engineering students.
4. Recommendations
In light of the findings of the study, it is recommended that the derived formulas be applied in solving problems in differential equation and advanced engineering mathematics. It is also recommended that the integrals of other hyperbolic functions such as hyperbolic tangent, cotangent, secant and cosecant be studied.
References
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[2]. D. Zill, S. Wright and W. Wright, “Calculus:Early Transcendentals, Jones and Bartlett Learning, 2009. [3]. E. Yomba, “Generalized Hyperbolic Functions to
Find Soliton – like Solutions for a System of Coupled Nonlinear Schrödinger Equations”, Physics Letters A, Vol. 372, No. 10, 2008.
[4]. P. Yusuf and U. Halime, “New Generalized Hyperbolic Functions to Find New Exact Solutions of the Nonlinear Partial Differential Equations”, Hindawi Journal of Mathemtics, Volume 2013, Article No. 201276, 2013.
[5]. I. Zucker, “The Summation of Series of Hyperbolic Functions”, Society for Industrial and Applied Mathematics, SIAM Journal on Mathematical Analysis, Vol. 10, No. 1, 2012.
