Suspicious Integrator
What has happened with my online integrator?
Age
16 to 18
Challenge level
Problem
I asked my integrating application to work out this integral:
Integrate[1000000/(10 + (1000000 - 10)/2^(4*x)), x]
I was given the answer
(25000*Log[-32*(99999 + 16^x)])/Log[2]
Try to evaluate this expression at $x=1$. What has gone wrong? Can you work out the actual real answer and verify by differentiation?
Did you know ... ?
Technology is undoubtedly a very useful mathematical tool, but needs to be used carefully and with some skepticism. If you use a calculating aid it is very important to check that the answer makes sense. Algebraic tools tend not to make errors, but often introduce unnecessary complexity into an answer.
Technology is undoubtedly a very useful mathematical tool, but needs to be used carefully and with some skepticism. If you use a calculating aid it is very important to check that the answer makes sense. Algebraic tools tend not to make errors, but often introduce unnecessary complexity into an answer.
Student Solutions
The integrator (with proper formatting) gave the answer for my integral $I$ as
$$I = \frac{25000\log[-32(99999 + 16^x)]}{\log 2}$$
The problem is that the logarithm of a negative number is complex. However, the rules of logarithms still apply, so we see that
$$
I = \frac{25000}{\log 2}\Big(\log(-32) + \log(99999+16^x)\Big)
$$
The $\log(-32)$ part, although complex, is a constant of integration which has been chosen by the integrator. Thus, the integral is of the form
$$
I = \frac{25000\log(99999+16^x)}{\log 2} + c
$$
This is perfectly real for real choices of $c$ and direct computation shows that this differentiates down to our starting function.