Dido525 2 years ago Evaluate the Intergal

1. Dido525

$\int\limits_{}^{}\frac{ 2x^4-4x^3+13x^2-6x+10 }{ (x-2)(x^2-2x+5)^2 }dx$

2. Dido525

I solved it using a beastly amount of partial fractions after solving a system of 5 equations. Please tell me there is another easier way.

3. DHASHNI

jus divide the numerator by x-2

4. Dido525

Why though? I have no reason to do long division.

5. DHASHNI

it is jus to simplify the question

6. Dido525

Hmm Okay... Let me see...

7. calmat01

Yep. it simplifies, reducing your amount of work.

8. calmat01

ok, maybe not. I really need a break. Making too many mistakes today. See you all later.

9. Dido525

Erm... THat doesent help O_o . I get: $2x^3+13x+20 +\frac{ 50 }{ 2x^3+13x+20 }$

10. satellite73

no it doesn't simplify but it has been cooked so that the partial fractions are nice integers

11. DHASHNI

yeah ^ he's right

12. Dido525

Ohh they are nice and all but even evaluating the damn partial fractions takes forever.

13. satellite73

there are some snap methods for getting some of them i think eliasaab who i have not seen here for a while had a snappy way of doing it

14. Dido525

So after the decomposition I get: $\int\limits_{}^{}\frac{ 2 }{ x-2 }+\frac{ 4 }{ x^2-2x+5 }+\frac{ x }{ (x^2-2x+5)^2 }dx$

15. Dido525

The first term is fine. It's the other two that take forever to integrate.

16. satellite73

all integration is a useless pain in the butt that is why they invented computers it is not math, it is wasted doodling

17. Dido525

Wolfram made I think... 9 substitutions. I refuse to do that.

18. TuringTest

@satellite73 D: blasphemy, integration is an art form!

19. satellite73

art who?

20. satellite73

waste a bunch of time showing off, saying "oh look, can find a function whose derivative is this, and a function whose derivative is that" when the truth is that if you pick a function out of a hat the probability you can find a nice closed for for the anti derivative is zero!

21. Dido525

I mean I got the correct answer. But isn't there an easier way to do this?

22. satellite73

23. Dido525

Damn it. Thanks anyways guys :)

24. TuringTest

haha, true that. PF is just plain annoying