Integrating a Simple Piecewise Function

Question

If I define

f[x_]:=If[x<0,-1,1]

and then integrate,say

Integrate[f[x],{x,-3,7}]

I get what I expect, namely 4.

But if I do the same thing with a different function

f[x_] := If[x < -1, 1/(2x^2), If[x > 1, 1/(2x^2), 0]]

and if I then integrate

Integrate[f[x], {x, -3, 7}]

I do not, as expected, get 16/21; instead I just get an expression consisting of an integral sign with -3 and 7 at the limits, followed by the definition of f just as I typed it above (with all the If statements, etc) and then a dx.

I can of course break the integral into two parts, either of which Mathematica handles perfectly well. But how do I get it to evaluate this expression without my manual intervention?

Answer 1

You can use Piecewise to define your piecewise functions. For example your second example could be defined as follows:

f[x_] := Piecewise[{{1/(2 x^2), Abs[x] > 1}, {0, True}}]

Integration:

Integrate[f[x], {x, -3, 7}]

yields 16/21

Answer 2

In Mathematica 9.0 the second integral evaluates to 16/21...