Different representations of the first order Bessel functions using maple

The aim of this paper is to find a closed form of the integrals ∫ 0π cos( x sin( t ) − nt ) d t for n = 0, 1, 2, … using the Maple computer algebra system. Although Maple 10 is not capable to calculate these integrals in one step, it turns out to be a very useful tool to solve this and similar kind of complex mathematical problems. During the problem solving process Maple proves that it is useful and, what is more, it is an indispensable partner. Maple helps us to formulate our conjecture, acts as an advisor and, last but not least, performs complex symbolic calculation instead of us.