## help_with_math 3 years ago Integration help! integral of sec^5 x dx

1. Walleye

2. across

Integration by parts!

3. across

4. help_with_math

yes integration by parts

5. Roachie

(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. Walleye

red formula will work if you have it

8. help_with_math

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

9. Roachie

sec^2x = 1 + tan ^2

10. Roachie

( 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. help_with_math

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. Roachie

derive tan x = sec^2x

14. Roachie

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

15. help_with_math

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

16. Roachie

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

17. help_with_math

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

18. help_with_math

or wat to do?

19. Roachie

hang on...

20. Roachie

im a little rusty

21. help_with_math

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

22. Roachie

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. Roachie

this IS a reduction formula for powers of sec.

24. help_with_math

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

25. help_with_math

then again do u mind tellin me wat that is

26. help_with_math

or how to use it

27. Roachie

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

28. help_with_math

cool i will prob understand that if u explain

29. help_with_math

the trig solution i mean

30. Roachie

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. Roachie

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

32. help_with_math

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

33. Roachie

a=1

34. help_with_math

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. help_with_math

where do u go on from there

36. help_with_math

omg i havent done definite integrals!

37. Roachie

no, I mistyped

38. Roachie

∫sec^5dx=1/4sec^3xtanx+3/4∫sec^3xdx

39. Roachie

no definite integrals here

40. help_with_math

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

41. Roachie

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

42. help_with_math

im confused how u got the previous step thou

43. Roachie

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

44. help_with_math

i mean teh 1/4 sec part im confused about

45. Roachie

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

46. help_with_math

ok so now how did u get 3/4?

47. Roachie

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

48. Roachie

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

49. help_with_math

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

50. Roachie

yea the trigonometric integrals take practice( obviously )

51. help_with_math

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. Roachie

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

53. Roachie

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

54. help_with_math

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

55. Roachie

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

56. help_with_math

wait but isnt is half secant

57. help_with_math

so how is teh integral jsut sec

58. Roachie

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

59. Roachie

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

60. help_with_math

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

61. Roachie

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. help_with_math

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

63. help_with_math

SRY TAHTS SUPPOOSED TO be ln not tn

64. Roachie

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. help_with_math

let me check now

66. Roachie

"table of integrals"

67. help_with_math

under wat heading shud i looj for te hreduction formulas

68. help_with_math

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. Roachie

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

70. help_with_math

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

71. help_with_math

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

72. Roachie

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

73. Roachie

there might be one for ∫sec^n (ax)dx

74. Roachie

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

75. help_with_math

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. help_with_math

i guess tahts anoterh form of te one u showed me

77. Roachie

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. help_with_math

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

79. Roachie

peace

80. help_with_math

ii imma go study sum more

81. help_with_math

gnite