# Find the indefinite integral using Integration by Partial Fractions: S8/(x^5 - 5x^2)  dxI am unsure what to do with the extra exponents.  I know that it needs broken down into 2 fractions A/? +...

Find the indefinite integral using Integration by Partial Fractions: S8/(x^5 - 5x^2)  dx

I am unsure what to do with the extra exponents.  I know that it needs broken down into 2 fractions A/? + B/?  and then you solve for A and B and then solve the problem, but I am unsure how to get the equations for the problem.

any help would be appreciated.

Thanks

tiburtius | High School Teacher | (Level 2) Educator

Posted on

It is a bit more complicated than that. It needs to be broken into 4 fractions this way:

First you find all real roots: `x_(1,2)=0,` `x_3=5^(1/3).` There are also 2 complex solutions of equation `x^2+5^(1/3)+5^(2/3)`. Now you can write

`1/(x^5-5x^2)=1/(x^2(x-5^(1/3))( x^2+5^(1/3)+5^(2/3) ))=`

`A/x + B/x^2 + C/(x-5^(1/3)) + (Dx + E)/( x^2+5^(1/3)+5^(2/3))`

To clearify you have `A` and `B` because 0 is double root `C` is for the third root and `Dx + E` are for 2 complex roots. If you add all the fractions you get the following equation.

`Ax(x-5^(1/3) ( x^2+5^(1/3)+5^(2/3) )) `

`+ B(x-5^(1/3) ( x^2+5^(1/3)+5^(2/3)))`

`+ Cx^2 ( x^2+5^(1/3)+5^(2/3) )`

`+ (Dx+E)x^2(x-5^(1/3)) = 1`

Now for `x=0` you get `B=-1/5`, for `x=5^(1/3)` you get

`C=1/(3 cdot 5^(4/3))` etc.

I hope this helps. For futher explanation consult literature for integrating rational functions (Bronstein, Demidovic, Apsen, ...).

You can also check links below. First link is to very usefull book about integration techniques (of course you don't have to buy the book, you e.g. can go to libary) and second link is for integrating rational functions.

Sources: