Question

Hey! I need help Evaluating the integral of [5*(x^3)-7x+4]/[(x^2)*(x-1)] dx using partial fractions...

Answer 1

It's traditional to use division first to reduce the numerator to some degree less than the degree of the denominator.

Answer 2

is this a long division problem, my original denominator was \[den=(x^{3}-x^{2})\] but that is linear, isn't it?

Answer 3

Both numerator and denominator are degree 3.

Answer 4

wouldn't I go straight over to partial fractions?

Answer 5

Your life will be better if you resist that temptation. Division will give you at least one trivial term for your integrand. Why would you not want that?

Answer 6

Long division gives \(5 + \dfrac{5x^{2} - 7x + 4}{x^2(x-1)}\).

See how nice and easy that '5' out front looks?!

Answer 7

that 5 does look pretty nice...let me run through the long division to see if I have any problems on that

Answer 8

You sound converted. Sweet!!

Answer 9

that was definitely the best response I've seen all week

Answer 10

and I got the division correct, sweet!

Answer 11

Don't tell anyone, but we can actually have a little fun doing mathematics.

Okay, now off to the Partial Fractions. What are your denominators?

Answer 12

\[x, x^{2},(x-1)\]
I believe that should be the decomposition

Answer 13

You have this. Pop it out and let's see what you get.

Answer 14

Oh...that's why I flipped my biscuits...the way I would have done it would have had a cubic against a quadratic.

I do believe I will get:
\[\int\limits_{}^{}\frac{5x^2-7x+4}{x^3-x^2} dx=\int\limits_{}^{}\frac{3}{x}dx+\int\limits_{}^{}\frac{-4}{x^{2}}dx+\int\limits_{}^{}\frac{2}{(x-1)}dx\]

which yields
\[3\ln(\left|x\right|) + \frac{4}{x} + 2\ln(\left| x-1 \right|) + C\]

Answer 15

Except for one thing. Something else is entirely missing. What is it?

Answer 16

+5 xD

Answer 17

That's it. We put it out there to be all pretty. No sense forgetting about it.

Excellent work.

Answer 18

I got a no-go from mathlab....are there any alternative forms

Answer 19

Perhaps I mistook your xD. That should be "5x".

Answer 20

that deserves at least three awkard turtles. Thank you for the help!

Answer 21

Just a tiny glitch. Now we have it.