I'm doing something wrong and would appreciate help finding the mistake. The last three integrations don't make sense. Ther's a note in Integrate[] documentation that says, "Substituting limits into an indefinite integral may not give the correct result for a definite integral," but doesn't say why or what to do about it.

a = Integrate[PDF[StudentTDistribution[1699], x], {x, -Infinity, 0}];
b = Integrate[PDF[StudentTDistribution[1699], x], {x, 0, Infinity}];
c = Integrate[
   PDF[StudentTDistribution[1699], x], {x, -Infinity, Infinity}];
d = Integrate[
   PDF[StudentTDistribution[1699], x], {x, -Infinity, -1.58989}];
e = Integrate[
   PDF[StudentTDistribution[1699], x], {x, -1.58989, 1.58989}];
f = Integrate[
   PDF[StudentTDistribution[1699], x], {x, 1.58989, Infinity}];
{a, b, c, d, e, f}
Show[
 RegionPlot[-1.58989 < x < 1.58989 && 
   y < PDF[StudentTDistribution[1699], x], {x, -2.5, 2.5}, {y, 
   0, .4}], 
 Plot[PDF[StudentTDistribution[1699], x], {x, -3, 3}, 
  PlotRange -> All, PlotStyle -> Red], AspectRatio -> 1/2]
Clear[a,b,c,d,e,f]