#### 6.72 Assume(16.2.01)

I am sure that this has been reported before.

> assume(-3<t);additionally(t<3);
> about(t);
Originally t, renamed t~:
is assumed to be: RealRange(Open(-3),Open(3))
> int(2/(t^2-9),t);
> int(-2/(9-t^2),t);
> -int(2/(9-t^2),t);

In each case the solution is `1/3ln(t~-3)-1/3ln(t~+3)`

. Maple does not recognize the
domain of the integrand at all and we have the log of a negative number. Ugly.

If you allow for the constant of integration to be complex, then the result makes
sense. If you evaluate this anitiderivative between real limits, you will get a real
answer.

I do realize, however, that this is diﬃcult to explain to a beginning calculus student when
you’re trying to teach them Maple.

From the online help of ‘int’ the statement:

Note that no constant of integration appears in the result.

Therefore the integral of an real valued integrand can have a constant imaginary
part;

> restart;
> assume(-3<t, t<3);
> int(2/(t^2-9),t):
> evalc(%):
> J := %;
J := 1/3 ln(3 - t) - 1/3 ln(t + 3) + 1/3 I Pi
> Jr := Re(J);
Jr := 1/3 ln(3 - t) - 1/3 ln(t + 3)

For `-3 < t < 3`

this is a real valued expression.

Try:

> f := int(2/(x^2-9),x=0..t);

and Maple will recognize the assumption made on t.