# [Maxima] New definite integrals and new problems

Dieter Kaiser drdieterkaiser at web.de
Sat Jun 6 15:26:14 CDT 2009

```Because of the last commits to limit.lisp we not only get nice new
limits for special functions, many thanks to Dan Gildea for this
extensions, but new results for interesting definite integrals too.

But because defint was not designed to integrate new special functions,
we get new problems too.

This is nice new and correct result:

(%i54) integrate(expintegral_ci(x),x,0,1);
(%o54) expintegral_ci(1)-sin(1)

But this is wrong:

(%i55) integrate(expintegral_ci(x),x,-1,1);
(%o55) -2*sin(1)+expintegral_ci(1)+expintegral_ci(-1)

defint does not know that the Exponential Integral Ci for a negative
argument is complex and has no method to look at singularities for new
functions.

By the way: It would be nice to get the result for the integral

integrate(exp(-x)*log(x),x,0,inf)

When we implement the new method simp-domain-error for the function
gamma_incomplete we get:

(%i58) integrate(exp(-x)*log(x),x,0,inf);
(%o58) -'limit(-%e^-x*log(x)-gamma_incomplete(0,x),x,0,plus)

The last step is to get the limit of the above expression which is known
to be -%gamma.

I am interested in this integral, because on the German wikipedia.de
there is a comment that Maxima is not even able to solve this integral.
Now, we have nearly implemented this type of integral.

Dieter Kaiser

```