## logarc

In the back of my calculus book there is a table of famous integrals.  Here’s integral number 21 in that table:

From Maxima integrate(), I get

What’s going on?

Both forms give a workable antiderivative for the original integrand:

Furthermore, we believe that both forms are correct because of this helpful identity for hyperbolic sine:

${\rm asinh}(z)=\ln(z+\sqrt{1+z^2}).$

Turns out (thanks to a Barton Willis for pointing me in the right direction) there’s a variable logarc that we can set to make Maxima return the logarithmic form instead of hyperbolic sine:

I haven’t yet encountered cases where this would be a bad idea in general, but I’ll update this if I do.

## logabs

In the first week of my differential equations course, we study methods of direct integration and separation of variables.  I like to emphasize that the absolute values can lend an extra degree of generality to solutions with antiderivatives of the form

$\int \frac{1}{u}\;du = \ln |u|+ C.$

As an example, for the initial value problem

$y' = \frac{x}{1-x^2}$,  $y(0)=1$,

it is convenient for treating all possible initial conditions ($x \ne \pm 1$) in one step to use the antiderivative

$y = -\frac{1}{2} \ln | 1-x^2 | + C.$

However, Maxima omits the absolute values.

For this case, we could consider only the needed interval $-1, but still…

Turns out we can set the Maxima variable logabs to make integrate() include absolute values in cases like this:

But then later in the course, I saw that logabs also impacts the Ordinary Differential Equation solver ode2().  I encountered an example for which Maxima, in particular solve() applied to expressions involving absolute value,  didn’t do what I wanted with logabs:true

For the logistic equation

$\frac{dP}{dt} = kP\left ( 1-\frac{P}{P_c} \right )$,  $P(0)=P_0$

we expect that by separating variables we can obtain the solution

$P(t) = \frac{P_0 P_c}{(P_c-P_0)e^{-kt}+ P_0}.$

Here’s what happens when we use ode2() with and without logabs:true: