Notes on 2010 problem 3a integral.

We see that Cos + Sin looks like a scaled shifted sine or cosine with the same 2 Pi period

Plot[Cos[t]+Sin[t], {t, 0, 2 Pi}, Ticks -> {{0, Pi/2, Pi, 3 Pi/2, 2 Pi}, {-1, 0, 1}}]

and we can show that this sum can be written as

FullSimplify[Sqrt[2] Cos[t - Pi/4]]
FullSimplify[Sqrt[2] Cos[t + Pi/4]]

So our integral takes the form

Clear[a, b, c]
Integrate[w E^(I  b w(Cos[t] )), {w, 0, a}, {t, 0, 2 Pi}]
Integrate[w E^(I   w(b Cos[t] + c Sin[t])), {w, 0, a}, {t, 0, 2 Pi}]
Plot[{Sinc[x], BesselJ[1, -x]/(-x)}, {x, -4 Pi, 4 Pi}]

(a^2) π Hypergeometric0F1Regularized[2,-(1/4) (a^2) ((b^2)+(c^2))] // TraditionalForm
 Manipulate[Plot[ x^2 Hypergeometric0F1Regularized[2,- arg x^2/4]
, {x, -4 Pi, 4 Pi}
, PlotRange->Full
, AxesLabel->{x, x^2 Hypergeometric0F1Regularized[2,- arg x^2/4]}
] , { {arg, 0.0001}, 0.01, 1} ]

π (a^2) (Hypergeometric0F1Regularized[2, -(1/4) (a^2) ((b^2)+(c^2))]) /. a -> R

π (R^2) Hypergeometric0F1Regularized[2,-(1/4) ((b^2)+(c^2)) (R^2)] /. b -> -k α

π (R^2) Hypergeometric0F1Regularized[2,-(1/4) (R^2) ((c^2)+(k^2) (α^2))] /. c -> -k β

π (R^2) Hypergeometric0F1Regularized[2,-(1/4) (R^2) ((k^2) (α^2)+(k^2) (β^2))] // Simplify

π (R^2) Hypergeometric0F1Regularized[2,-(1/4) (k^2) (R^2) ((α^2)+(β^2))] // TraditionalForm