## anonymous one year ago how to do this integral?

1. anonymous

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

2. misty1212

HI!!

3. anonymous

hello!!

4. misty1212

how are you at partial fractions?

5. anonymous

i understand it a bit (split into parts, right? A/B?) but don't know quite how to apply it!

6. misty1212

it is going to suck want to do it step by step, or cheat?

7. anonymous

can you show me how to do it step by step?

8. misty1212

$\frac{A}{x-3}+\frac{B}{x-2}+\frac{C}{(x-2)^2}$

9. anonymous

okay! what next? :O

10. misty1212

that means $A(x-2)^2+B(x-3)(x-2)+C(x-3)=x^2+1$

11. misty1212

let me know if that is clear, then we can continue

12. anonymous

yes:)

13. anonymous

it is clear:) what happens next?

14. misty1212

we can get C right away by putting $$x=2$$ because the A and B will drop out

15. misty1212

if you put $$x=2$$ you get $C(2-3)=2^2+1$ or $-C=5$ making $$C=-5$$

16. misty1212

got that?

17. anonymous

yes:) sorry, OS connection was very poor!! what happens next?

18. IrishBoy123

get A by setting x = 3 same technique

19. anonymous

okay, so then i get 3^2 + 1 = 10 ?

20. anonymous

A=10?

21. IrishBoy123

$A(3-2)^2+B(3-3)(3-2)+C(3-3)=3^2+1$ $\checkmark$

22. anonymous

ok!! what do i do next?

23. IrishBoy123

$10(x-2)^2+B(x-3)(x-2)-5(x-3)=x^2+1$ you need B so set x to anything really. but let's just go with x = 0.

24. anonymous

okay! so B=1?

25. IrishBoy123

sure? 10(4) + B(-3)(-2) - 5(-3) = 1 6B = (1 -15 -40)????

26. anonymous

ohh oops!! it equals -55!

27. anonymous

-0.1111 = B?

28. anonymous

@IrishBoy123 i have to leave for an appointment now, but will be back in a few hours!! is it okay if we continue then, or if i come back and see what you say?

29. IrishBoy123

$\color{red}6B = -5 \color{red}4$

30. IrishBoy123

sure

31. anonymous

32. anonymous

so B = -9 right? what happens next?

33. IrishBoy123

let's agree where we got to first

34. IrishBoy123

$\frac{10}{x-3}-\frac{9}{x-2}-\frac{5}{(x-2)^2}$ are we sure that's a good summary?

35. anonymous

okay! so A=10, B=-9 and C=-5 ?

36. IrishBoy123

now you must do $\int \frac{10}{x-3}dx- \int \frac{9}{x-2}dx-\int\frac{5}{(x-2)^2} dx$ 3 separate integrals

37. anonymous

ooh okay, how do i start off?

38. IrishBoy123

first one $\int \frac{10}{x-3}dx$

39. anonymous

okay, so we get 10lnx-3 ?

40. IrishBoy123

y second?

41. anonymous

so it is not that?

42. IrishBoy123

$\int \frac{9}{x-2}dx$

43. anonymous

or it is?

44. IrishBoy123

first part is right, do the second now

45. anonymous

ohhh i see now and so now second we get 9lnx-2 ?

46. anonymous

and for the third, we get -5/(x-2) ?

47. IrishBoy123

yes. just make a note that we need to put the correct signs between the integrals righ at the end as it is ∫dx - ∫dx - ∫dx last one $\int\frac{5}{(x-2)^2} dx$

48. IrishBoy123

yes put it all together now

49. IrishBoy123

don't bother writing it out if you are happy with it all

50. anonymous

$\int\limits_{}10lnx-3 dx - \int\limits_{}9lnx-2dx + \int\limits_{} \frac{ 5 }{ x-2 }dx$

51. anonymous

like that?

52. IrishBoy123

no, drop all the integration signs now that you have integrated so $10 \ln(x-3) - 9\ln(x-2) + \frac{ 5 }{ x-2 }$

53. IrishBoy123

OK?

54. anonymous

ohh okay!! what now? :)

55. IrishBoy123

i think the fat lady is now singing.

56. anonymous

hahaha ohh so we are done?

57. anonymous

@IrishBoy123 ?

58. IrishBoy123

OMG!!!! we forgot something....

59. anonymous

what?? :O

60. IrishBoy123

the integration constant!!

61. IrishBoy123

how embarassing after all that :-))

62. anonymous

ohh + C ?

63. IrishBoy123

indeed!

64. anonymous

ohh okie!! yay!! thank you so much!!

65. IrishBoy123

you and misty did all the heavy lifting, we just finished it off:) good night!

66. anonymous

good night!:D