Hey y'all,
Can someone please explain what I'm doing wrong, I don't really understand what the absolute function is and those two lines, and why they switch the signs in the answer
Thanks
I want to make this clear - I'm not a fan of answering an already answered question. I think it's rude to the people who have already put in an answer - particularly well explained and clear answers like the ones above. However, I feel that while the responses from Bri and james clearly explain what the modulus function is, they don't explain WHY it appears when integrating these kinds of functions.
To understand that, let's instead think about what integration is - it's an inverse operation of differentiation. So, we know that (-1/3)ln(4-3x) is an antiderivative (not necessarily the only one, but it's definitely one of them) of 1/(4-3x) because if you differentiate (-1/3)ln(4-3x), you get 1/(4-3x). In fact, let's do that working out:
Noting the application of the chain rule in the second step. Okay, so this leaves us with:
Which must mean that:
Well, for fun, what about when we differentiate (-1/3)ln(3x-4)? Well, as above:
Well hang on a second, this means that:
And that must mean that:
But, we already decided that the integral of 1/(4-3x) was equal to the ln(4-3x), not ln(3x-4)! This seems like a contradiction, but it's not - in truth, it's because the integral of 1/(4-3x) could be EITHER of the logarithmic functions, we just don't know which one, because it depends on whether 3x-4 is positive, or if 4-3x is positive. So, to say that it could be either one and we just don't know which, we put the modulus signs in there - because as other said, those modulus signs will MAKE the inside positive, so it will always work. Then, we can remove those modulus signs later if we know the initial conditions - but that's getting into specialist territory, and to memory you're not expected to know about removing the modulus signs in methods.
Hopefully this makes a bit more sense conceptually, instead of just being another "random" rule that you have to remember.