Question

Please help me with this one, I'm struggling with it.
Integral of 1 / x^2.(x+1)

Answer 1

partial decomp?

Answer 2

it splits into linear denominators ...
\[\frac{1}{x^2(x+1)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+1}\]

Answer 3

What if I put like this
= A/(x+1) + (Bx+C)/x^2

Answer 4

then its wrong ...

Answer 5

you have to consider all the possible ways that the decomp can occur

Answer 6

lol I guess. Thanks a lot

Answer 7

After calculating, i got A=0, B=1, C=0. does it sound right?

Answer 8

the x^2 denom always makes me uneasy :) it looks quadratic to me and I always want to put a linear form above it ... but for some reason its assume to be a multiplicity of a single linear and a constant above it is stated in the texts

Answer 9

i dont get those values ... can you show your work?

Answer 10

ok. When I redo it, I got:
1 = Ax(x+1) + B(x+1) + Cx^2
1 = Ax^2 + Ax + Bx + B + Cx^2
1 = x^2(A+C) + x(A+B) + B
=>
B=1
A + B= 0 => A=-1
A + C = 0 => C = 1

Answer 11

there are 2 methods that I recall, a coverup and a rework. the coverup is fine for some cases but not all cases so I just go with the rework .. try to rebuild how you would add them:

\[\frac{1}{x^2(x+1)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+1}\]
\[\frac{1}{x^2(x+1)}=\frac{Ax(x+1)}{x~x(x+1)}+\frac{B(x+1)}{x^2(x+1)}+\frac{Cx^2}{x^2(x+1)}\]
\[\frac{1}{x^2(x+1)}=\frac{Ax(x+1)+B(x+1)+Cx^2}{x^2(x+1)}\]
these are only equal if the parts are equal to one another giving us the tops to compare
\[1=Ax(x+1)+B(x+1)+Cx^2\]
lets zero out the terms

when x=0 we are left with:\[1=B(0+1)\]

when x=-1 we are left with:\[1=C(-1)^2\]

knowing B and C we can assume any value for x to assess the A, lets use something simple like x=1:
\[1=A(1)(1+1)+1(1+1)+1(1)^2\]

Answer 12

A = -3/2 ... if im seeing it right

Answer 13

@amistre64, the decompositions are the same
\[\frac{A}{x+1}+\frac{Bx+C}{x^2}=\frac{A}{x+1}+\frac{B}{x}+\frac{C}{x^2}\]

Answer 14

yeah, hartnn pointed that out to me in a different question :) i either had a bad textbook, or a truamatic memory lol