## anilorap 4 years ago i want to confirm the answer of... integrate lnx/x dx

1. anilorap

my answer is ln lxl +c

2. myininaya

let u=lnx du=dx/x $\int\limits_{}^{}u du=\frac{u^2}{2}+C=\frac{(lnx)^2}{2}+C$ we can even check this: $(\frac{(lnx)^2}{2}+C)'=2*\frac{lnx}{2}*\frac{1}{x}+0=\frac{lnx}{x}$

3. myininaya

do you got it?

4. anilorap

yes

5. myininaya

k :)

6. anilorap

q. how do i know what to choose?

7. myininaya

i was looking for $\int\limits_{}^{}f(x)*f'(x)dx$

8. myininaya

sometimes its not always that easy though

9. myininaya

someone said it is like an art and i agree

10. anilorap

jajaj..ok

11. myininaya

it is not always easy to know what substititon to make but since it was in the form of above it was easy let u=f(x) du=f'(x) dx

12. myininaya

$\int\limits_{}^{}\frac{lnx}{x}dx=\int\limits_{}^{}lnx*\frac{1}{x} dx$

13. myininaya

so it was easy for me to notice pretty fast that (lnx)'=1/x so thats why i let u=lnx

14. anilorap

ok..perfect

15. anilorap

so how can i integrate tanx dx..... even though i know thats it is equal to -lnlcosxl+c

16. myininaya

$\int\limits_{}^{}\frac{sinx}{cosx} dx$ right?

17. myininaya

tanx=sinx/cosx agree?

18. anilorap

ohhhhhhhh... ok

19. anilorap

yes

20. anilorap

so that du/u

21. myininaya

now what do you think the substitution will be?

22. anilorap

with the negative sign of the sin

23. anilorap

u=cos

24. anilorap

du=sin x

25. anilorap

sorry du=-sinx

26. myininaya

27. anilorap

oh yea

28. myininaya

so show me what that gives us don't integrate yet

29. anilorap

then u have integral of du/u

30. myininaya

negative or positive?

31. anilorap

- integral of du/u

32. myininaya

yes and now integrate

33. anilorap

so my result will be -ln/u/+c

34. anilorap

-ln/cosx/+c

35. myininaya

yes :)

36. anilorap

yeaaa thank uuuu

37. myininaya

np

38. myininaya

so whats this $\int\limits_{}^{}-tanx dx$ equal to?

39. anilorap

ln/cosx/+c

40. anilorap

becuase the negative sign will change to postive once u take the derivative of cosx

41. anilorap

why u pic u =sinx?

42. myininaya

accident

43. myininaya

u=cosx du=-sinx dx

44. anilorap

:)

45. anilorap

good....:

46. myininaya

you want to try one i made up? its not hard

47. anilorap

ok

48. myininaya

$\int\limits_{}^{}\frac{x+1}{x^2+2x}dx$

49. anilorap

ok.. let me try

50. anilorap

u=x^2 + 2x du=2x+2 dx =2(x+1) dx so, 1/2du=x+1 dx

51. myininaya

very good so far

52. anilorap

1/2 integral of du/u =1/2 lnlul+c =1/2 ln/x^2+2xl+c

53. myininaya

:)

54. anilorap

yeaaaaaaaaaaaaa

55. anilorap

i love this stuff

56. myininaya

lol it is nice

57. myininaya

$\int\limits_{}^{}\frac{f'(x)}{f(x)}dx$ so when we have this what will out answer be in the form of?

58. anilorap

f(x)*f'(x)

59. myininaya

look at the one you just did didn't we have f'/f dx?

60. anilorap

ohh... sorry.. yes yes

61. myininaya

=ln|f(x)|+C

62. anilorap

yes

63. myininaya

$\int\limits_{}^{}[f(x)]^n*f'(x)dx$ $n \neq -1$ what about this? what will our answer be?

64. myininaya

i will give you a hint let u=f(x) so du=f'(x) dx

65. anilorap

jjajjjaj... nice hint

66. myininaya

where are you from? you say jajaja alot lol

67. anilorap

dominican republic

68. anilorap

i am just laughing

69. myininaya

jajaaja= laughing?

70. anilorap

yess... how can i laugh??

71. myininaya

its fine the way you type your laughter no worries i like it

72. anilorap

lol

73. myininaya

so this all makes a little more sense?

74. myininaya

the integral stuff?

75. anilorap

why?

76. myininaya

im asking do you get the integral stuff better?

77. anilorap

ohhhhhh,, yess

78. myininaya

good :) im fixing to go to sleep goodnight

79. anilorap

for me is mornig.. but good nigth,,,

80. myininaya

good morning* i didn't sleep all night lol

81. anilorap

jajajajaaja

82. anilorap

ayayay.. no good

83. anilorap

thank u very much...

84. myininaya

np if you want to try another this is a good one $\int\limits_{}^{}x*\sqrt{x+1} dx$ the answer is given above in one of the other threads if you want to check yourself

85. myininaya

later

86. anilorap

ok,, thanks