## anonymous 4 years ago can someone help me solve an integral?

1. anonymous

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2. Akshay_Budhkar

the thing that immediately comes to my mind is use the $\int\limits_{}^{} uvdx$ formula twice

3. anonymous

ok in the midst of doing that

4. Akshay_Budhkar

but that will be a lengthy business, must have a shorter route

5. anonymous

so what do u suggest?

6. anonymous

cos(x)= Re[e^(i x)] x= e^(ln x) $\int e^{ln(x)} e^x e^{i x} dx$

7. anonymous

$\int e^{ln(x)+x+i x} dx$

8. Akshay_Budhkar

Complex numbers!! y didnt i think of that!

9. anonymous

huh what is e^ix?

10. anonymous

cos(x)= Re[e^(i x)] what is this step?

11. anonymous

Using integration by parts three times to solve it isn't that bad either. It would take a while to type out all of the equations, but if you use 1. u=xe^x, dv=cos(x), then 2. u=xe^x, dv=sin(x), then 3. u=e^x, dv=cos(x) over your three stages, you will get the answer of $\int xe^x\cos x dx = \frac{1}{2}xe^x(\sin x + \cos x) - \frac{1}{2}e^x\sin x + C$

12. anonymous

Eh. I would hesitate using complex numbers if you don't know how, you can get yourself into trouble.... using the natural extension of integration by parts to three functions is simple and straightforward, and makes sense already.

13. Akshay_Budhkar

@imperialist that will be a bit of lengthy business though, if she knows complex numbers it will be faster

14. anonymous

@Jemurray3, I agree. It would take far longer to explain why you can use complex numbers here (and even longer to understand it!) than 3 applications of integration by parts.

15. anonymous

Thanks guys :D

16. anonymous

I am just gonna wait to see what myin has to say for herself

17. myininaya

First lets evaluate: $\int\limits_{}^{}e^x \cos(x) dx$ $=e^x \cos(x)-\int\limits_{}^{}e^x(-\sin(x)) dx=e^x \cos(x)+\int\limits_{}^{}e^x \sin(x) dx$ $=e^x \cos(x)+(e^x \sin(x)-\int\limits_{}^{}e^x \cos(x) dx)$ So we have $\int\limits_{}^{}e^x \cos(x) dx=e^x \cos(x)+(e^x \sin(x)-\int\limits_{}^{}e^x \cos(x) dx)$ So we shall solve this for: $\int\limits_{}^{}e^x \cos(x) dx$ So we have $\int\limits_{}^{}e^x \cos(x) dx=\frac{1}{2}e^x \cos(x)+\frac{1}{2}e^{x}\sin(x)+c_1$ ok now let's look back at $\int\limits_{}^{} x e^x \cos(x) dx$ Apply integration by parts again! :) $=(\frac{1}{2}e^x \cos(x)+\frac{1}{2}e^x \sin(x))x-\frac{1}{2}\int\limits_{}^{}(e^x \cos(x)+e^x \sin(x)) dx$ So we still need to find $\int\limits_{}^{}e^x \sin(x) dx$ So we will need integration by parts again $=e^x \sin(x)-\int\limits_{}^{}e^x \cos(x) dx$ $=e^x \sin(x)-(e^x \cos(x)-\int\limits_{}^{}e^x (-\sin(x)) dx)$ So we need to solve this for: $\int\limits_{}^{}e^x \sin(x) dx$ $\int\limits_{}^{}e^x \sin(x) dx=\frac{1}{2} e^x \sin(x)-\frac{1}{2}e^x \cos(x)+ c_2$ So we need to put all of this together! lol

18. myininaya

$\int\limits_{}^{}x e^x \cos(x) dx=\frac{1}{2}x e^x \cos(x)+\frac{1}{2}x e^x \sin(x)-\frac{1}{2}(\frac{1}{2}e^x \cos(x)+\frac{1}{2}e^x \sin(x)+$ $\frac{1}{2}e^x \sin(x)-\frac{1}{2}e^x \cos(x))+C$ after adding up all the constants I get C lol because the sum of some constants is still constant I think I got every part in there

19. anonymous

LOL WOW

20. anonymous

Thanks myin but u did it differently then what the others suggested

21. anonymous

I wish I could give you a dozen medals myininaya for typing all of that up! :)

22. Akshay_Budhkar

i tell her that for every answer! she is more than cool! :D

23. Akshay_Budhkar

just tell ur kids not to trouble me @ myin :P :P

24. anonymous

Thanks myin imperialist and jemurray:D I will have to review th steps now :D

25. myininaya

Great question.

26. myininaya

You guys are really sweet! :)

27. Akshay_Budhkar

ur kids are sweet too :D

28. Akshay_Budhkar

so are you :D

29. anonymous

Oh myin u deserve the medal not me LOL

30. myininaya

I don't have kids lol

31. myininaya

you have kids not me lol

32. anonymous

Akshay is like1o years younger than you LOL

33. Akshay_Budhkar

yea i told her that she doesnt get me! her kidds come to my bakery and dont pay me

34. anonymous

LOL and eat ur trash

35. Akshay_Budhkar

LOL junk to be precise

36. myininaya

lol

37. Akshay_Budhkar

myin is blushing when i told her she has kids :P

38. anonymous

lol she is dating satellite LOL hehe

39. Akshay_Budhkar

!

40. Akshay_Budhkar

satellite is 40 years old!

41. myininaya

You guys are crazy. I never met satellite in person.

42. anonymous

lol i was teasing

43. myininaya

i know

44. Akshay_Budhkar

rld study dont tease :P :P

45. anonymous

yup got loads of work to do. Thanks myin U r awesome

46. Akshay_Budhkar

47. anonymous

LOL dont even have to say that

48. Akshay_Budhkar

yea yea~ :P i m so cool :P

49. anonymous

if u live in my city = cool

50. Akshay_Budhkar

later=D

51. anonymous

ya bye guys

52. anonymous

myin helped me solve it but i know u like integrals. The second I posted it u disappeared LOL Too bad

53. anonymous

My bad, I'll help on the next one.

54. anonymous

ya but now i am doing stupis things. I gotta refer to the tables of integrands and solve accordingly. Soon I will be solving using trig subsitution so i will be back :D