Honestly speaking, this kind of question does not have a definite answer. Whenever you are trying to do such integration, first thing you should check is whether your function is convergent or not. Considering your function, it will not converge for any parameter values.
So I have to give a generic answer. If you don't know the properties of your function, evaluate it on the parameter space and then plot it. You might get an idea of the parameter dependence of the integral.
For your example
f[d_, k_, t0_, u_] := t0/Pi Exp[-u^d Cos[d Pi/2]] Cos[u^d Sin[d Pi/2] - k t0 u]
Does not converge for d=1
NIntegrate[ f[1, 1, 1, u], {u, 0, Infinity}]
$2.731938882417951*10^{27949}$
So choose smaller d
and evaluate it
dat1 = Table[{k, t0,
NIntegrate[
f[.1, k, t0, u], {u, 0, Infinity}]}
,{k, -0.5, .5, .1}, {t0, 0.,1., .1}];
dat2 = Table[{k, t0,
NIntegrate[
f[.5, k, t0, u], {u, 0, Infinity}]}
,{k, -0.5, .5, .1}, {t0, 0.,1., .1}];
dat1 = Flatten[dat1, 1];
dat2 = Flatten[dat2, 1];
ListPlot3D[dat1, PlotLabel -> "d=0.1"]
ListPlot3D[dat2, PlotLabel -> "d=0.5"]

For advanced integration techniques you might be interested in this.
Now let say you want the functional relationship. So you can use Interpolation
.
g = Interpolation[dat2]
Plot3D[g[x, y], {x, -0.5, 0.5}, {y, 0, 1}]

g[k,t0]
is your approximate solution for d=0.5
.