lanvins said:
For
Q1, the book has it as -3loge |2x-5|+c, where the 2x is positve and the 5 is negative. why is that?
Q2. i don't get the point of this step
y' = 6u^-2.du.(dx/du)
dx/du = -1/2
Q3. same problem as question 1, the book has it as y = -1/3 log|-4+3x| +C
Q4. the back of the said the answer is 1/2loge |x|+1
Thanks for helping me by the way
Question 4: The problem we have here is our basic log principles.
Everyone has agreed so far that y=1/2ln2x+c, yes?
Sub in the point (e^2, 2)
2=1/2(ln2e^2)+c
Let us remember that this is the logarithm of 2(e^2)
Thus, we cannot turn it into 2ln2e.
We can only do that if we it is ln(2e)^2, agreed?
So:
1/2(ln(2xe^2)+c=2
1/2(ln2+lne^2)+c=2
1/2(ln2+2lne)+c=2
1/2(ln2+2x1)+c=2
1/2ln2+1/2x2+c=2
1/2ln2+c+1=2
c=1-1/2ln2
Sub the value of c back in to get:
y=1/2ln2x+(1-1/2ln2)
y=1/2(ln2x-ln2)+1
y=1/2(ln(2x/2))+1
y=1/2lnx+1
It has already been explained that the absolute value is regarding the doman of the function. Just be careful with logs - make sure that in ln(f(x)^n), the power is for the whole of f(x) before you move it to the front.
Oh and in regards to question 2, there is the 'reverse chain rule' which is not really a reverse chain rule as it only works for linear functions. i.e. when integrating f(x)^n, f(x) must be linear (the greatest power of x is 1). I can't remember this rule right now, but yeah it should be in your textbook.