1. When it comes to the constant 'c' in Integrated functions, if we had to divide both sides by two or something like that, would the 'c' remain as 'c' (which my teacher says is the answer) or is it, 2c?
So what does that constant 'c' represent? We add in that constant of integration to represent the multitude of different functions that have that same derivative (the family of curves).
Let's take the example of
(chosen an example like this for a particular reason which we'll see later)
But
is just a constant, so we can reduce the answer to the indefinite integral to:
Elaborating a bit from that, if we took the function:
 = \frac{4}{3}\log_e{(3x+2)} - \frac{4}{3}\log_e{3} + c)
and differentiated it, we would get
(using chain rule)
 = \frac{4}{3x+2})
To summarise:
If we had
 = \frac{4}{3}\log_e{(3x+2)} + c)
, we'd get the same derivative. The same goes for
 = \frac{4}{3}\log_e{(3x+2)} + 2c)
. Or if c was equal to 123, we get that same derivative for
 = \frac{4}{3}\log_e{(3x+2)} + 123)
and
 = \frac{4}{3}\log_e{(3x+2)} + 246)
2. √(x^2 ) does that equal, x or |x|?
over all real numbers, but for positive real numbers, than
is true.
We can just sub in two numbers to see this I guess,
^2} = \sqrt{4} = 2)
and
and
and
and
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