Help me integrate this please :)

Thanks, I don't think those types of partial fractions are in the c4 syllabus but it seems like the most sensible option

Only one more, if you don't mind...

x^2/X^3 + 1

Anyway, there's a simple rule I've discovered (I'm pretty sure it's in a textbook somewhere, but it works all the same). If the numerator has a coefficient with a power of 1, and the denominator has a coefficient with a power of 1, n-d = 0, which means that one of the factors is just a constant, i.e $A$.

If the numerator has a power of 2 and the denominator has a power of 1, then one of the factors will be a polynomial of order 1, i.e, $Ax + B$.

If the numerator has a power of 3 and the denominator has a power of 1, then one of the factors will be a polynomial of order 2 i.e, $Ax^2 + Bx + C$.

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This doesn't make sense, coefficient of what? A coefficient is the scalar multiplier of any term involving a variable - so saying the "coefficient of the numerator is 1" doesn't really have any meaning.

I don't know why you've decided to shoot me down, but I'll explain:

Partial fractions we break down are the reverse of using either + or - operations on Algebraic fractions. What you're doing when you're using a partial fraction is decomposing it into it's initial fractions that make up the full fraction.

All I was saying, was by using the method I've stated, you can skip directly using long division and just treat the whole thing as you would a normal fraction you were decomposing. Whereas if you were using LD, you would be directly looking for the values of the unknowns, which could potentially be a time waster, and also are very tricky to manipulate in some situations.

In the example I gave, that would mean that after putting it in the terms I gave, you simply multiply through by the denominator on the left hand side, giving $A(x^2+4x) + B(x+4) + C(x)$.

I wasn't trying to say that I was doing anything groundbreaking, all I was saying that was in order to use the method of partial fraction decomposition that people doing A Level Maths are familiar with, you can treat an improper fraction in the way I said, shortening it to 1 step and making integration etc quicker and cleaner.

That's all fine, I was wondering why you brought it up when the method you just explained was the method that was being encouraged throughout the thread anyway. I thought you were on to something that I had missed in my explanations or something, or a shortcut, so I wanted to know what you meant but that doesn't seem to be the case.

Nope. I wasn't trying to do anything special. The OP just said that he hadn't seen partial fractions done like that before, so I was just explaining to him what was going on when they were being done in your examples.