Integrating cos(sin(theta))

Posted on 2008-09-23 12:12 By tsunamiBTP (3)

Categories:

Math Advice (Student)

I have a function when expressed in polar coordinates such that a trig function resides inside a trig function. In calculus 101 we all learned that integrating the product or quotient of 2 or more trig functions requires integration by parts but I have never run across the case where a trig function is a function of another trig function. Any one have any references I should consult on to learn how to handle this?

non-elementary

Comment on 2008-09-23 14:00 from Robert Israel ( Maple Rating 4

1555)

Maple returns int(cos(sin(theta)),theta) unevaluated. Since this is the "transcendental" case, for which I believe Maple's implementation of the Risch algorithm is pretty much complete, I'm pretty confident that Maple is correct in that this function has no elementary antiderivative.

definite or indefinite?

Comment on 2008-09-26 09:31 from edgar ( Maple Rating 2

186)

Certain definite integrals of this kind are known...

int(cos(a*sin(t)),t=0..Pi/2);

(1/2)*Pi*BesselJ(0, a)
---
G A Edgar

However

Comment on 2008-09-26 11:20 from jakubi ( Maple Rating 4

943)

this knowledge is somewhat limited:

int(cos(a*sin(t)),t=0..Pi/2);
                         1/2 Pi BesselJ(0, a)

int(cos(a*sin(t)),t=0..Pi);

                           Pi BesselJ(0, a)

int(cos(a*sin(t)),t=0..2*Pi);

                          2 Pi
                         /
                        |
                        |      cos(a sin(t)) dt
                        |
                       /
                         0

Apparently, the periodicity and symmetries of the integrand are not used in the latter case.

periodicity and symmetries

Comment on 2008-09-26 14:54 from jpmay ( Maple Rating 2

170)

Many of the internal integrators do check for periodicity and symmetries, but apparently they are not catching compositions of trig functions. I will look into beefing this up a bit.

John