Super quick integral C4

When integrating x/x+1

why is it wrong to rewrite that as 1+x

When integrating x/x+1

why is it wrong to rewrite that as 1+x

When integrating x/x+1

why is it wrong to rewrite that as 1+x

When integrating x/x+1

why is it wrong to rewrite that as 1+x

What you have written, $\frac{x}{x}+1$ is equal to 2.

What you probably meant, $\frac{x}{x+1}=\frac{x+1-1}{x+1}=1-\frac{1}{x+1}$.

You can easily check that $\frac{a}{b+c} \ne \frac{a}{b}+\frac{a}{c}$.

How did you get this

$\frac{x}{x+1}=\frac{x+1-1}{x+1}=1-\frac{1}{x+1}$.

That is done my intuition and very clever thinking, a standard trick that people doing STEP for example become familiar with. Alternatively, you could just use the substitution:

$u=1+x$

Which works as-well.

Could you explain how you did that division though? Because the mark scheme does the same thing

He has added 0 in a fancy way.

Take his fraction:

$\dfrac{x}{x+1}[br][br]1-1=0[br][br]therefore: \dfrac{x+1-1}{x+1} = \dfrac{x}{x+1}[br][br]$

we can split the numerator, NOT the denominator

$[br][br]\dfrac{x+1-1}{x+1} = \dfrac{x+1}{x+1} - \dfrac{1}{x+1} = 1 - \dfrac{1}{x+1}[br][br]\int 1 - \dfrac{1}{x+1}\ dx = x - ln|x+1| + c[br][br]$

That made perfect sense, thank you so much! You explained it really well!

Just going a tad bit of topic but I'm also looking to prepare for STEP and do you know where I can pick up on "tricks/techniques" such as that?

That made perfect sense, thank you so much! You explained it really well!

Just going a tad bit of topic but I'm also looking to prepare for STEP and do you know where I can pick up on "tricks/techniques" such as that?

Only place you pick up tips and tricks is just doing paper, after paper, after paper, after paper I am afraid.

However, if you are just getting started on STEP, here is a very useful document written by the chief examiner.

http://www.maths.cam.ac.uk/undergrad/admissions/step/advpcm.pdf

There is also another booklet called advanced problems in mathematics, do not touch that book until you are confident in answering questions of this linked books difficulty. I would recommend in this order, assuming you re doing all 3 STEP papers, you can cut off this chain wherever it applies to you.

Advanced Problems in Core Mathematics -> STEP 1 Papers -> Advanced Problems in Mathematics/STEP 2 Papers -> STEP 3 Papers.

Ah yes, I've come across that pdf by Stephen Siklos - I even saw him at Cambridge!

But when you began preparing for STEP, how long after (roughly) did you start to think "yeah I'm starting to get the hang of this"?

1 month ago , it took me a really long time.

Don't blame you! How long have you been preparing then?