Define the integration operator I by
so I f is an antiderivative of f.
Define the second antiderivative I2 by applying I to f twice:
It turns out that
To see this, notice that both expressions for I2 are equal when x = a, and they have the same derivative, so they must be equal everywhere.
Now define I3 by applying I to I2 and so forth defining In for all positive integers n. Then one can show that the nth antiderivative is given by
The right side of this equation makes sense for non-integer values of n, and so we can take it as a definition of fractional integrals when n is not an integer.
Related post: Fractional derivatives