## cherrilyn 5 years ago evaluate the integral

1. cherrilyn

$\int\limits_{?}^{?}(x ^{2} + 3x -44)dx/(x+3)(x+5)(3x-2)$

2. amistre64

partial fraction decomp it looks like; unless we can factor the top to cancel some bottoms

3. amistre64

x^2 +3x -44 = A(x+5)(3x-2) ; x = -3 x^2 +3x -44 = B(x+3)(3x-2) ; x = -5 x^2 +3x -44 = C(x+3)(x+5) ; x = 2/3

4. cherrilyn

woohoo thats what I got. but now how do I find the actual values of the letters?

5. amistre64

-44 = A(2)(-11) -44/-22=A A = 2 ---------------------- 25 -59 = B(-2)(-17) -34 = B(34) B = -1 x^2 +3x -44 = C(x+3)(x+5) ; x = 2/3

6. amistre64

when x = a number that zeros the other 2; use it and solve for that letter

7. cherrilyn

but sometimes it leaves behind more than one variable

8. amistre64

x^2 +3x -44 = C(x+3)(x+5) ; x = 2/3 (2/3)^2 +3(2/3) - 44 = C(2/3 +3)(2/3 +5)

9. amistre64

sometimes it does, but use it first to get some solids :) with linears on the bottom, its not a problem

10. cherrilyn

okay. for the last equation I got 4/9 - 42 = c(2)(2/3 + 5)

11. amistre64

(4/9) +2 - 44 = C(11/3)(17/3) (4/9) - 42 = C(11/3)(17/3)

12. amistre64

-374/9 = C(187/9) C = -374/189 reduce as wanted

13. cherrilyn

wow how do you work so fast?

14. amistre64

i use my head and mess up alot lol

15. cherrilyn

lol oh... do you think you can help me with another similar problem?

16. amistre64

perhaps; but let me get thru this one first :)

17. cherrilyn

okay! thanks for all of your help :)

18. amistre64

we did C correctly right? that monster fraction scares me into thinking I messed it up lol

19. amistre64

i typoed the 187 into a 189 ;)

20. amistre64

C = 2 :)

21. amistre64

-2

22. cherrilyn

wait, in this problem.. there is no Dx+E?

23. amistre64

so our integral becomes: [S] 2/(x+3) - 1/(x+5) - 2/(3x-2) dx

24. amistre64

the bottom is already factored for us into linear forms; linear form dont need the Dx+E form

25. cherrilyn

ahh okay! Yayyyy. :D

26. amistre64

if we had an irreducible quadratic in the denom; then we would use the Dx+E form above it to aid us

27. cherrilyn

like.. (x-1)^3?

28. amistre64

nope; that is a multiple of a linear and still only needs a A B C single atop it

29. cherrilyn

really? how would it look.. A/x-1 + B/x-1 + c/x-1?

30. amistre64

x^2 + 10 is irreducible quadratic; cant be factored to real linears

31. amistre64

A/(x-1) + B/(x-1)^2 + C/(x-1)^3

32. cherrilyn

ohh.. hmm . and that would be... Ax+B/x^2+10?

33. amistre64

our integral has become: 2ln(x+3) - ln(x+5) -2ln(3x-2) +C

34. cherrilyn

35. amistre64

thats the form we would use above it yes :)

36. amistre64

we could probably turn this into one ln(yadayada)+C if we want, but this is good :)

37. cherrilyn

wow thank you!!!!

38. amistre64

youre welcome :)