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Weekly Challenge 24: Suspicious Integrator

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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.