Question

express in partial fraction
1/x(a-x)

Answer 1

-A/x - B/x-a=1/x(a-x)
-A(x-a)-B(x)=1
-Ax+Aa-Bx=1
Collect powers
-A-B=0
+Aa=1
A=1/a
-1/a-B=0
B=-1/a

Answer 2

so its 1/ax-1/a(x-a) but iam not sure if this is correct or not

Answer 3

Im curious as to how you arrived at negative signs?

Answer 4

\[\int\limits_{}^{}\frac{ 1 }{ x(a-x) }\]

Using partial fractions this will become
\[\frac{ A }{ x }+ \frac{ B }{ (a-x) } = \frac{ 1 }{ x(a-x)}\]

Answer 5

Now we would cross multiply
\[\frac{ A }{ x }+\frac{ B }{ (a-x) } = A(a-x)+B(x)\]
Since the bottom terms are both the same we have
\[A(a-x) +B(x) = 1\]

Answer 6

Although it is stated "express in partial fraction" so we may not be asked to solve for the partial fraction yet it is good practice to know know.

Answer 7

You can see if it is write by combining the fractions @Abmon98
Test:

\[\frac{1}{ax}-\frac{1}{a(x-a)}=\frac{1(x-a)}{ax(x-a)}-\frac{1(x)}{ax(x-a)}=\frac{x-a-x}{ax(x-a)}=\frac{-a}{ax(x-a)}\\ =\frac{-1}{x(x-a)}=\frac{1}{x(a-x)}\]
Therefore the decomposition you have works
:)
Great job.

Answer 8

right not write*

Answer 9

Thank you @myininaya and @Johnbc for your help :D