Two integrals have been developed which are needed to solve certain problems of mathematical physics related to the Bessel and Euler equations. Integrals of this type have known application in problems using cylindrical coordinates

Rostafinski, W. A.

English

NASA Technical Reports Server

May 1972	 B72-10225 
•
	 NASA TECH BRIEF 
Lewis Research Center 
NASA Tech Briefs announce now technology derived from the U.S. space program. They are issued to encourage 
commercial application. Tech Briefs are available on a subscription basis from the National Technical Information 
Service, Springfield, Virginia 22151. Requests for individual copies or questions relating to the Tech Brief program may 
be directed to the Technology Utilization Office, NASA, Code KT, Washington, D.C. 20546. 
Indefinite Integrals of Products of Some 
Exponential and Trigonometric Functions 
Two integrals have been developed which are needed 
to solve certain problems of mathematical physics related 
to the Bessel and Euler equations. These integrals are: 
f
Pcos
xp+i 
 (b In x) dx=
(p+ 1)2+b2 
I(p+ 1) sin (b in x)—b cos (b In x)] +c 
where x is the variable, b is an arbitrary constant, and p 
is unrestricted. Tables of integrals contain these integrals 
for pO and p= — 1, but not for an unrestricted value of 
the exponent. 
To obtain the above integrals, consider 
fx Peib In xdx f ib+pdx_ ib+p+1 +c (p1-I) 1-ib
where i—_,./:i, e is the base of the natural logarithm, and 
c is the integration constant. 
Rationalizing the denominator and expressing the 
exponential function in terms of the circular functions 
results in:
Ixe in xdX= [(p+i)_ib] xP+i J	 (p+1)2+b2 
[cos (b in x) + i sin (b In x)] 1-c 
Splitting into the real and the imaginary parts yields the 
two integrals. 
Notes: 
i. Integrals of this type have known application in 
problems using cylindrical coordinates. 
2. Documentation may be obtained from: 
Technology Utilization Officer 
Lewis Research Center 
21000 Brookpark Road 
Cleveland, Ohio 44135 
Reference: B72-10225 
Patent status: 
No patent action is contemplated by NASA. 
Source: W. Rostafinski
Lewis Research Center
(LEW. 1 1493) 
[b sin (b In x) + (p+ 1) cos (b In x)]+c; 
o
IxPsin
xp+
 (b In x) dx= 
(p+ 1) 2+b2 
•
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Indefinite integrals of products of some exponential and trigonometric functions

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