## anonymous 4 years ago Integration help! integral of sec^5 x dx

1. anonymous

2. across

Integration by parts!

3. across

4. anonymous

yes integration by parts

5. anonymous

(sec^2x)(sec^2)(sec x)

6. across

I read it as sec^5(x)x for a moment there. Hmm, in this case, you have to use a reduction formula.

7. anonymous

red formula will work if you have it

8. anonymous

well i havent learned taht so is tehre anoterh way to do it by parts

9. anonymous

sec^2x = 1 + tan ^2

10. anonymous

( 1 + tan ^2 x ) ( 1 + tan^2 x ) (sec x )

11. across

I had to look it up$\frac{\sin(x)\sec^{m-1}(x)}{m-1}+\frac{m+2}{m-1}\int\sec^{m-2}(x)dx$

12. anonymous

this is wat i have so far: $\int\limits_{}^{}\sec^3 x (\sec^2 dx)$ $\int\limits_{}^{}\sec^3 x (1+\tan^2 x dx)$ $\int\limits_{}^{}(\sec^3 x +\sec^3 x \tan^2 x )dx$

13. anonymous

derive tan x = sec^2x

14. anonymous

so do a u sub with u = tan x , du = sec^2 x dx

15. anonymous

$\int\limits_{}^{}(\sec^3 x) + \int\limits_{}^{}(\sec^3 x \tan^2 x) dx$ can i use u sub after this step

16. anonymous

you will prolly end up with a ( 1 + u ) ^(to some power ) then it a simple integration

17. anonymous

can u show me the steps after the one i just did

18. anonymous

or wat to do?

19. anonymous

hang on...

20. anonymous

im a little rusty

21. anonymous

ughh i hate problems liek this!! XD...requires too much work

22. anonymous

yea, it may have to be broken up into parts I cant get rid of one of the sec^x check out: http://www.wolframalpha.com/input/?i=integrate+sec^5+x

23. anonymous

this IS a reduction formula for powers of sec.

24. anonymous

i did n idont understand it b.c it uses smthng i didnt leard which is teh reduction formula thing

25. anonymous

then again do u mind tellin me wat that is

26. anonymous

or how to use it

27. anonymous

yea, your instructor may not accept a reduction formula answer, it the quick way, but there is a trigonometric solution for this.

28. anonymous

cool i will prob understand that if u explain

29. anonymous

the trig solution i mean

30. anonymous

there is an identity that will lead to an integral of u. anywa here is a like with the reduction formula for secant, and others http://www.sosmath.com/calculus/integration/moretrigpower/moretrigpower.html

31. anonymous

as a matter of fact I bet your textbook has a table of integrals for integrals of powers of secant

32. anonymous

what is the dffnce between sex^n (x) and sec^n (ax)

33. anonymous

a=1

34. anonymous

i wud use the first formula right? sec^5 (x) = sec ^5-2 (x) sec^2(x) = sec^3 (tan^2 x * sec^2 x -1)

35. anonymous

where do u go on from there

36. anonymous

omg i havent done definite integrals!

37. anonymous

no, I mistyped

38. anonymous

∫sec^5dx=1/4sec^3xtanx+3/4∫sec^3xdx

39. anonymous

no definite integrals here

40. anonymous

hmm how did u get 1/4 sec ...etc

41. anonymous

then apply the reduction formula ( or integrate ) $\int\limits \sec ^ 3 x dx$

42. anonymous

im confused how u got the previous step thou

43. anonymous

its from the reduction formula, it ends with an integral of ∫sec^3xdx

44. anonymous

i mean teh 1/4 sec part im confused about

45. anonymous

in the reduction formula it 1/(a(n-1) n=5, the power of secant a = 1

46. anonymous

ok so now how did u get 3/4?

47. anonymous

from the reduction formula: (n-2)/(n-1) again n = 5, the power of the secant

48. anonymous

you instructor will be either impressed or put off that you are using the integral tables. :)

49. anonymous

lol...he will b impressed hopefully...i have a test 2mm n im trying to udnerstand probs idk

50. anonymous

yea the trigonometric integrals take practice( obviously )

51. anonymous

srry this is new to me so after taht step wat do u do..u still have a integral of sec^3 dx to deal with

52. anonymous

right! so integrate the sec^x or use the formula again, but this time with n = 3, see?

53. anonymous

sec^x is short, but it indefinite integral has many parts.

54. anonymous

so after using n=3 do u get 1/2 sec x dx

55. anonymous

right ( n-2)/(n-1) this time the formula will end with $\int\limits \sec x dx$

56. anonymous

wait but isnt is half secant

57. anonymous

so how is teh integral jsut sec

58. anonymous

no the formula will end ith ∫secxdx, integrate that and you are done.

59. anonymous

$\int\limits \sec x dx = \ln ( \sec x +tanx ) + C$

60. anonymous

so teh final answer is: 1/4 sec^3 x tanx + 3/4 ...idk teh rest

61. anonymous

1/4 sec^3 x tanx + 3/4∫sec^3xdx then apply the formula again for ∫sec^3xdx that answer will end with ∫sec xdx integrate ∫secxdx, done!

62. anonymous

so the final asnwer is: 1/4 sec^3 x tanx + 3/8 sec x tanx + tn (secx+tanx) + C

63. anonymous

SRY TAHTS SUPPOOSED TO be ln not tn

64. anonymous

right, distribute the 3/4, I didnt work it all out but it looks right. Those tables are in you textbook right, in the appendix?

65. anonymous

let me check now

66. anonymous

"table of integrals"

67. anonymous

under wat heading shud i looj for te hreduction formulas

68. anonymous

i see basic forms, forms invlving sqrt of (a^2+u^2, forms involving (a^2 - u^2), forms involvung sqrt of (u^2 - a^2), forms involving a+bu, trig forms, inverse trig forms, exp and log forms, hyperbolic forms

69. anonymous

Tables of integrals, Lists of Integrals usually in the appendix, could be on the inside covers.

70. anonymous

yeah i see them in the appenxix...jsut under wat heading of integrals will i fimd teh reduction formulas

71. anonymous

or wher u jsut letting me know taht tehre was a table of integrtals

72. anonymous

that really what we were using, an integration formula, so you can integrate by parts or look up the formula.

73. anonymous

there might be one for ∫sec^n (ax)dx

74. anonymous

might not http://integral-table.com/

75. anonymous

ths is wat is in my bk...sec^n du = 1/n tan u sec^n-2 + (n-2/n-1) *integral of sec^n-2 u du

76. anonymous

i guess tahts anoterh form of te one u showed me

77. anonymous

right that one dont account for a coefficient on x. so, yea, you seem to have it. Im out. ( gotta brush up on my integrals ;) )

78. anonymous

THANK U SOOOOO MUCH!!! PLUS THE WEBSITE LOOKS SUPER HELPFUL =)

79. anonymous

peace

80. anonymous

ii imma go study sum more

81. anonymous

gnite