Question

Guys what is partial fraction decomposition?

Answer 1

\[\frac{1}{x(x+1)}=\frac{a}{x}+\frac{b}{x+1}\]for some a,b

Answer 2

this is an example of partial fraction decomposition of 1/(x(x+1))

Answer 3

so what is the aim of a partial fraction decomposition?

Answer 4

Useful in particular, when integrating.
It's usually easier to integrate when the denominator-polynomial is only a linear one.

It just results in a logarithm.

Answer 5

...unless your context isn't calculus D:

If so, then... sorry ^_^

Answer 6

lets figure out what a,b are.
multiply everything by x(x+1)
\[1=a(x+1)+bx=ax+a+bx=x(a+b)+a\]here is the "trick
we have x(a+b)+a = 1
since we have no x's on the RHS we know that a+b=0 and since the rhs has only numbers we know that a=1
so a=1, and b =-1
so
\[\frac{1}{x(x+1)}=\frac{1}{x}-\frac{1}{x+1}\]

Answer 7

As @terenzreignz said, I would rather integrate the RHS and not the LHS

Answer 8

There are always exceptions, though :)
Like...
\[\Large \int \frac{1}{1-x^2}dx\]

Answer 9

But... yeah, that's the long and short of it... sometimes it's easier to deal with sums of simpler fractions than just one 'big' fraction (big here meaning a rather complicated polynomial for a denominator)

Answer 10

thank you guys ^.^