Evaluate CDF of Student-t distribution
BBC BASIC, 155 ASCII characters, tokenised filesize 140
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  a=1s=.000003INPUTx,v
  IFv&lt;1v=9999
  FORw=1TOv-1a=w/a
  NEXTb=SGN(x)*s*a/SQR(v)IFv MOD2b=b/PI ELSEb=b/2
  f=.5FORz=0TOABS(x)STEPs
  f+=b/(1+z*z/v)^(v/2+.5)NEXTPRINTf

The code is in two parts. The first calculates the coefficient that is independent of x. and stores it in b. The algorithm is a bit like a continued fraction, but a mathematician would probably call it a simple quotient of two products. It's based on a formula from the Wikipedia page in the question, which is similar to a factorial but has the product (v-1)(v-3)... divided by the product (v-2)(v-4)... This is possible because we are only interested in half-integral values of the gamma function (arbitrary values of the gamma function would be a lot harder.) Using this formula we never have to calculate either gamma function, so we avoid large numbers. The largest intermediate value is approximately the square root of v.
The second is a straightforward integration from 0 to x by summation. It is known that f(0) is 0.5 so we start from there and work outwards. the sign and magnitude of b are modified before the loop, in order to be able to give the final answer directly.
Pretty version, ungolfed
This contains a few extra lines to add colour and to plot the data. The range x=0 to endpoint, and f(x)=0.5 to 1 is shown. Also v is modified to 2^v to be able to show the full range of v values without exceeding the limits on colour. In order to speed the drawing up, there are two less digits in s than in the golfed version. Even so, there is only one discrepancy when compared with http://www.wolframalpha.com/input/?i=student+t-distribution+probability: 0.8130575 rounds up to 0.8131. The golfed version give the correct answer (0.8130) for this and all the other cases below.
  a=1                              :REM accumulator for gamma series
  s=.0001                          :REM step size for x integration
  INPUTx,v
  IFv&lt;1 v=9999                     :REM if v&lt;1, set v to a value that gives results indistinguishable from infinity
  GCOLv                            :REM pick a colour
  v=2^v                            :REM for the pretty version, modify v to 2^v
  FORw=1TOv-1
    a=w/a                          :REM calculate series as per Wikipedia
  NEXT
  b=a/SQR(v)                       :REM divide a by the necessary values...
  IFv MOD2 b=b/PI ELSE b=b/2       :REM to give the coefficient, per Wikipedia

  b*=SGN(x)*s                      :REM modify according to sign and step size
  f=.5                             :REM initialise f(x)=.5 at x=0
  FORz=0TO ABS(x) STEPs
    f+=b/(1+z*z/v)^(v/2+.5)        :REM perform integration as simple &quot;barchart&quot; sum
    PLOT69,z*100+200,(f-.5)*1600   :REM plot a point
  NEXT
  PRINT f