While trying to find an answer to this problem on the forum, I came across this integral:
$$ I=\int_c^\infty {\sin \big(x+{k \over x} \big) \over x}dx \tag 1$$
Where $c$ and $k$ are real numbers. I tried to solve this using Feynamn's trick. I defined a parameter $p$ such that:
$$\overline I (p)=\int_c^\infty {\sin \big(px+{k \over x} \big) \over x}dx \tag 2$$
Taking the derivative of $\overline I$ with respect to $p$ I get:
$$\overline I'=\int_c^\infty {\cos \big(px+{k \over x} \big)}dx \tag 3$$
To get rid of one constant in the cosine, I used the substitution $t=px$ and got:
$$\overline I'={1 \over p}\int_{pc}^\infty {\cos \big(t+{kp \over t} \big)}dt \tag 4$$
By grouping the constants as $cp = b$ and $kp = q$ we get:
$$\overline I'={1 \over p}\int_{b}^\infty {\cos \big(t+{q \over t} \big)}dt \tag 5$$
The next thing I did was use the Euler trigonometric identity to write:
$$\overline I'={1 \over 2p}\int_{b}^\infty \Bigg ( e^{i \big(t+{q \over t} \big)} + e^{-i \big(t+{q \over t} \big)} \Bigg ) dt \tag 6$$
Here I got stuck. I would appreciate some help in solving this problem.
