# Can one Integral have different solutions based on the algorithm used to solve it?

This is my first post here and I searched the internet for things like "can u substitution yield an invalid result when finding an antiderivative?" and "manual integration and maxima produce different results" but this didn't find any useful links, which is why I'm posting here.

# Background

I have a rather simple expression of which I want to find the indefinite integral/antiderivative of:

$$\frac{1}{y - \frac{115}{3}}$$

For context, this is part of a problem, where I am asked to solve the following ODE without using a SolveODE command:

$$\frac{dy}{dt} = -\frac{3}{50} \; \left(y - \frac{115}{3} \right)$$

Looking at the solution provided with the problem, I see that integrating the expression described at the beginning of this question should result in the following:

$$\ln \left( \left|x - \frac{115}{3}\right| \right) + C$$

However, when solving using GeoGebra CAS to calculate the Integral (my exact input was Integral(1 / (x - 115 / 3))), I get the following output:

$$\ln \left( \left|3 \; x - 115\right| \right) + C$$

# Using external help

Because I think that the antiderivative included in the official solution does not equal the one calculated by the Integral command in GeoGebra, I went to Integral-Calculator.com (the link leads to my calculation on the website) to find out what I did wrong.

I typed in my expression (1 / (y - 115 / 3)) and told it to solve for $$y$$.

# A surprising result

Integral-Calcularot.com found both of the solutions.
Its results section has two parts: One describing a "manual" solve (whatever that means) and one which displays the solution calculated by the Maxima software:

# Several questions arise

• How can there be two different (and conflicting) solutions to one math problem?
• What did I miss?
• Is one of the two solutions invalid?
• Are the two solutions actually equal?

# PS

AFTER typing my entire question, this website provided me with the following other (simular) question:

Different results when integrating 1/(x^2-9) with computer tools

I'm not sure if this would be considered a "duplicate" of the other one, but it doesn't help me understand my issue better; i.e. the two answers (or that question in and of itself) doesn't completely cover what I'm trying to do.

# Update

I've been testing and fiddling around a bit more and noticed the following:

It turns out flipping the fraction on its head and using $$^{-1}$$ leads to the desired behaviour. Any idea why?

• Consider that C is an unspecified constant and that $ln(ab)=ln(a)+ln(b).$
– Paul
May 15, 2020 at 14:46
• The "solution" of an indefinite integral is a set, that is, the expression $f(x)+C$ for arbitrary constant represent a set of solutions. The solution can be represented in various ways but the resulting set solution must be the same May 15, 2020 at 15:46
– Sil
May 15, 2020 at 18:14