1. ## Find primitive function

Hi

Is this 'supposed' to be solved with partial fractions decomposition?

Problem:
Find all primitive functions to:

$\int \frac{1}{x^{2}(x-1)^{3}} \; dx$

Kinda need some starting help if so is the case, thx!

2. Here is a CAS solution.

3. Hi

im sorry but it is $\frac{1}{x^{2}\cdot (x-1)^{3}}$ and not $\frac{1}{x^{3}\cdot (x-1)^{3}}$

But, should I do:

$\frac{1}{x^{2}\cdot (x-1)^{3}} = \frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{(x-1)}+\frac{D}{(x-1)^{2}} + \frac{E}{(x-1)^{3}}$ ?

4. Originally Posted by Twig
Hi

Is this 'supposed' to be solved with partial fractions decomposition?

Problem:
Find all primitive functions to:

$\int \frac{1}{x^{2}(x-1)^{3}} \; dx$

Kinda need some starting help if so is the case, thx!
Try and find constant $a - e$ such that

$
\frac{1}{x^{2}(x-1)^{3}} = \frac{a}{x} + \frac{b}{x^2} + \frac{c}{x-1} + \frac{d}{(x-1)^2} + \frac{e}{(x-1)^3}
$

Yes.

5. Originally Posted by Twig

Is this 'supposed' to be solved with partial fractions decomposition?
No.

Originally Posted by Twig

Problem:
Find all primitive functions to:

$\int \frac{1}{x^{2}(x-1)^{3}} \; dx$
Put $x=\frac1t$ and your integral becomes $\int{\frac{t^{3}}{(t-1)^{3}}\,dt}=\int{\frac{(t-1)^{3}+(t-1)^{2}+2t(t-1)+(t-1)+1}{(t-1)^{3}}\,dt},$

and I think you can integrate that.

6. Originally Posted by Krizalid
No.

Put $x=\frac1t$ and your integral becomes $\int{\frac{t^{3}}{(t-1)^{3}}\,dt}=\int{\frac{(t-1)^{3}+(t-1)^{2}+2t(t-1)+(t-1)+1}{(t-1)^{3}}\,dt},$

and I think you can integrate that.
That's a bit .... final!

There are several ways of doing it. None are 'right' or 'wrong'. Partial fractions is one way. And there's no doubt about the usefulness and ease of the reciprocal substitution, which is another way.

However, I suspect that whoever set this question would have had partial fractions in mind.