This is the last part of a question I was trying to do. Whats the best way to do this integration?

(x+2)/(x^2+2x+2)+(2-x)/(x^2-2x+2) dx
Thanks.

Kerwin

It looks very nice Kerwin and I am sure I could reduce that into its standard integrals.

But I am unsure of how you got it to the form you show me.

I can get it from the original form to the form below -

ò(x+2)/[(x+1)²+1]+(2-x)/[(x-1)²+1]
What do we do now to get it into the form you have it.

x+2 º (x+1)+1 º

1 2

(2x+2)+1

2-x º (1-x)+1 º -

1 2

(2x-2)+1

and complete the square on the denominators:

x²+2x+2 º (x²+2x+1)+1 º (x+1)²+1

x²-2x+2 º (x²-2x+1)+1 º (x-1)²+1

Kerwin

Thanks Kerwin.

I am not sure whether I could have deduced that from scratch. It is something we are expected to know and do? Or is there another way to do it?

The above method is one of a number of strategies for dealing with integration of algebraic fractions. Here's a list of things that can be helpful for different algebraic fractions.

• Recognising it as a standard integral eg of sin^-1 or tan^-1
• Observing that the numerator is (a multiple of) the derivative of the denominator: integral of f'(x)/f(x) dx = ln|f(x)|+c
• 1/(ax+b)ⁿ is straightforward by substituting u=ax+b
• Separating it into two fractions by breaking the numerator as above: you probably noticed that Kerwin chose how to do it to give himself nice easy fractions that could be integrated by the first two methods listed here
• Separating it into two (or more) fractions by using partial fractions.

It is nearly worth noting that the integral of (mx+n)/(ax² +bx+c) is

mln(a x2+b x+c) 2a + 2n-m b a   ______ Ö4b c-c2 tan-1 æ ç ç ç è 2a x+b   ______ Ö4b c-c2 ö ÷ ÷ ÷ ø

although I don't think anyone would commit this to memory.

Brad

Thanks for that Brad.
Though I doubt whether my memory can recall that with ease. But it's well worth taking note of. Thanks again