Help with C3 Integration question

Integrate x/x-2 in the range from 3 to 4.
So far I've only been taught how to integrate via substation so I thought u = x-2 and tried doing the usual but it doesn't seem to work out. Is there an alternative method?

Integrate x/x-2 in the range from 3 to 4.
So far I've only been taught how to integrate via substation so I thought u = x-2 and tried doing the usual but it doesn't seem to work out. Is there an alternative method?

You could also write the fraction x/(x-2) as 1 + 2/(x-2). If you subtract 2 and then add 2 to the numerator it might be easier to see why.

How did you get this?

How did you get this?

The substitution should work - post your working, and someone can check it out.

Alternatively, you can just perform the polynomial/synthetic division, if you've covered it (don't know what module that is), and then integrate.

So I wrote out the equation as x(x-2)^-1. And let u=x-2. Therefore du/dx = 1. dx/du = 1.

so x*u^-1. to get x in terms of u i did u+2=x
So (u+2)*u^-1. I multiplied out the bracket to get 1+2u^-1.
If this is integrated this is the same as u+2lnu. And then I put in the new ranges from intergral u=1 and u= 2. (originally x=3 and x=4) so 2+2ln2 - 1+2ln1 however this does not give me the correct answer of 1+2ln2. Does the 1 not need to be integrated into u? is that the mistake Im making here?

So I wrote out the equation as x(x-2)^-1. And let u=x-2. Therefore du/dx = 1. dx/du = 1.

so x*u^-1. to get x in terms of u i did u+2=x
So (u+2)*u^-1. I multiplied out the bracket to get 1+2u^-1.
If this is integrated this is the same as u+2lnu. And then I put in the new ranges from intergral u=1 and u= 2. (originally x=3 and x=4) so 2+2ln2 - 1+2ln1 however this does not give me the correct answer of 1+2ln2. Does the 1 not need to be integrated into u? is that the mistake Im making here?

2+2ln2 - 1+2ln1

1+2ln2

These are the same thing since ln(1) = 0.

For future questions please post all your working at the start so that we know where you're stuck.

Note: (a+b) - (c+d) = a + b - c - d, (note the bold emphasis).

What bold emphasis?