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Wednesday, May 6, 2020

Integral Trigonometri

Dalam melakukan integrasi pada fungsi trigonometri diperlukan teknik khusus agar proses integrasi lebih cepat. Hal ini karena terdapat beberapa identitas trigonometri.






Identitas Pythagorean
  1. sin2 x + cos2 x=1
  2. tan2 x  +1 = sec2 x
  3. cot2 x  +1 = csc2 x


Identitas Setengah Sudut
  1. sin2 x  = [1- cos 2x]/2
  2. cos2 x  = [1+ cos 2x]/2

Tipe-tipe Integrasi Trigonometri
  1. ∫ sinn x  dx  dan  ∫ cosn x  dx 
  2. ∫ sinm x  cosn x  dx 
  3. ∫ sin mx  cos nx  dx ;  ∫ sin mx  sin nx  dx ; dan ∫ cos mx  cos nx  dx 
  4. ∫ tann x  dx  dan  ∫ cotn x  dx 
  5. ∫ tanm x  secn x  dx  dan ∫ cotm x  cscn x  dx 

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46 comments:

  1. Sondang Venita Br. SamosirMay 6, 2020 at 6:27 AM

    Dalam melakukan integritas Trigonometri perlu teknik yang cepat dan terdapat beberapa identitas trigonometri yaitu: Identitas Pythagorean, Identitas Setengah sudut dan Tipe Tipe Integrasi Trigonometri

    ReplyDelete
  2. Losi eka putriMay 6, 2020 at 6:37 AM

    This comment has been removed by the author.

    ReplyDelete
  3. Losi eka putriMay 6, 2020 at 6:40 AM

    1. Integral sin^2 x dx=integral 1-cos 2x/2 dx
    =integral 1/2 dx-
    integral 1/2 cos 2x dx
    =1/2x - 1/4 sin 2x + C

    ReplyDelete
  4. Nova Asmeri DewiMay 6, 2020 at 6:43 AM

    This comment has been removed by the author.

    ReplyDelete
    Replies
    1. Nova Asmeri DewiMay 6, 2020 at 6:55 AM

      Bentuk integral trigonometri memiliki 5 tipe:
      1. ∫ sin^n x dx dan ∫ cos^n x dx
      2. ∫ sin^m x cos^n x dx
      3. ∫ sin mx cos nx dx ; ∫ sin mx sin nx dx ; dan ∫ cos mx cos nx dx
      4. ∫ tan^n x dx dan ∫ cot^n x dx
      5. ∫ tan^m x sec^n x dx dan ∫ cot^m x csc^n x dx

      Delete
  5. Nadia Putri WulandariMay 6, 2020 at 6:44 AM

    Saya akan mencoba menjawab soal latihan yang bapak kasih
    1. Integral (sin^2)x dx = (1/2)x-(1/4)sin 2x+C
    2. Integral (sin^3)x dx = cosx-(1/3)(cos^3)x+C

    ReplyDelete
  6. Shafira NurulitaMay 6, 2020 at 6:44 AM

    2. -cosx + 1/3 + cos³x + c
    5. 1/3 tan³x - tanx + x + c

    ReplyDelete
  7. Nova Asmeri DewiMay 6, 2020 at 6:51 AM

    This comment has been removed by the author.

    ReplyDelete
  8. Shafira NurulitaMay 6, 2020 at 6:58 AM

    4.  1/10 sin5x + 1/6 sin3x + c

    ReplyDelete
  9. Desnawati HutapeaMay 6, 2020 at 7:01 AM

    5. integral tan^4 x dx
    = integral tan² x tan² x dx
    = integral tan² x (sec²- 1) dx
    = integral tan² x sec² x - integral tan² x dx
    = 1/3 tan³ x - tan x + x + c



    ReplyDelete
  10. Elisabeth rajagukgukMay 6, 2020 at 7:05 AM

    1. 1/2x-1/4sin2x+c
    5. 1/3 tan³x-tanx+x+c

    ReplyDelete
  11. Desiwati br marbunMay 6, 2020 at 7:12 AM

    Latihan
    2. Integral Sin^3dx
    = integral sin^2x SinX
    = cos X-1/3cos^3x +c

    5. Integral tan^4 x dx
    = integral Tan^2 Tan^2 dx
    = 1/3 Tan^3x - Tanx + x + c

    ReplyDelete
  12. Dian AnggrainiMay 6, 2020 at 7:13 AM

    Saya akan menjawab soal latihan yang bapak berikan
    No 6. -2tan^-1/2 x+2/3 tan^3/2 x+c
    No 7. 2/3 sec^3/2 x+2 sec^-1/2 x+c

    ReplyDelete
  13. Diah maya pradestaMay 6, 2020 at 7:25 AM

    1) =1/2 x-1/4 sin 2x+c
    2) = -cos x + 1/3 cos^3 x +c

    ReplyDelete
  14. Zelly Permata SariMay 6, 2020 at 7:35 AM

    2. ∫sin³x dx
    Solusi :
    ∫sin³x dx = ∫sin x (1-cos² x) dx
    = ∫sin x dx - ∫sin x cos² x dx
    = -cos x + ⅓ cos³ x + C

    ReplyDelete
  15. Samudra rizqiMay 6, 2020 at 7:39 AM

    This comment has been removed by the author.

    ReplyDelete
  16. Samudra rizqiMay 6, 2020 at 7:41 AM

    2. In sin^3 x dx
    = In sin^2 x sin x dx
    = In (1-cos^2) sin x dx
    = In sin x - In sin x cos^2 x dx
    = - cos x + cos x^3/3 + c

    ReplyDelete
  17. Shofa Hanifah BahrilMay 6, 2020 at 7:45 AM

    4) (Sin 3x)/6 + (Sin 5x)/10 + C
    5) (Tan x^3)/3 - Tan x + x + C

    ReplyDelete
  18. erlina sihombingMay 6, 2020 at 7:52 AM

    This comment has been removed by the author.

    ReplyDelete
  19. Vivin AcnesyaMay 6, 2020 at 7:55 AM

    This comment has been removed by the author.

    ReplyDelete
  20. erlina sihombingMay 6, 2020 at 7:56 AM

    Dalam mempermudah penyelesaian integral trigonometri, maka dapat digunakan 3 bentuk identitas trigonometri, yaitu :
    1.Identitas Pythagorean
    2.Identitas Setengah Sudut
    dan menggunakan Tipe-tipe Integrasi Trigonometri

    latihan.
    3. integral sin^3 (3x).cos^4 (3x) dx
    misalkan u= cos 3x
    integral -sin^3 (3x).cos^4 (3x). 1/3 du
    integral -sin^3 (3x).u^4 (3x). 1/3 du
    integral -(1-c0s(3x)^2)u^4)/3 du
    integral (u^6-u^4 )/ 3
    =cos ^7 (3x)/21 - cos ^5 (3x) / 15 + C

    ReplyDelete
  21. Lucia Iqta NurfirzaMay 6, 2020 at 8:08 AM

    No. 4
    Integral [cos(x)cos(4x]dx= integral 1/2[cos(5x)+cos(-3x)]dx
    = integral 1/2[cos(5x)+cos(3x)]dx
    = 1/2 integral [cos(5x)+cos(3x)]dx
    = 1/2 integral cos(5x)dx+ integral cos(3x)dx
    = 1/2 [1/5 sin(5x) + 1/3 sin(3x)]
    = 1/10 sin(5x)+1/6 sin 3x +C

    ReplyDelete
  22. Vivin AcnesyaMay 6, 2020 at 8:09 AM

    Jawaban
    1)∫sin^2 x dx
    sin (x)^2 = (1-cos(2x))/2
    ∫sin^2 x dx
    =∫(1-cos (2x))/2 dx
    = 1/2 × ∫ 1 – cos (2x) dx
    = 1/2 × (∫ 1 dx - ∫cos⁡(2x) dx)
    = 1/2 × (x - ∫cos⁡(2x) dx)
    = 1/2 × (x - sin⁡(2x)/2) + C
    = 1/2x - sin⁡(2x)/4 + C

    3) ∫sin^3 3x cos^4 3x dx
    Misal :
    u = cos (3x)
    du/dx = - sin(3x) × 3
    du = -sin(3x) × 3 dx
    1/-sin(3x) × 3 du = dx
    ∫sin^3 3x cos^4 3x dx
    = ∫sin^3 3x cos^4 3x × 1/-sin(3x) × 3 du
    = ∫-sin^3 3x cos^4 3x × 1/sin(3x) × 3 du
    = ∫-(sin⁡(3x)^(2) cos⁡(3x)^(4))/3 du subs cos(3x) dengan u
    = ∫-sin⁡(3x)^(2)× u^4/3 du
    = ∫-1-cos(3x)^2 × u^4/3 du
    = ∫-(1-u^2) × u^4/3 du
    = ∫- u^4 - u^6/3 du
    = 1/3 × ∫u^6 - u^4 du
    = 1/3 × (u^7/7 - u^5/5) + C
    = 1/3 × (cos(3x)^7/7 - cos(3x)^5/5) + C
    = (cos(3x)^7/21 - cos(3x)^5/15) + C




    ∫▒/

    ReplyDelete
  23. Debora OktalitaMay 6, 2020 at 8:14 AM

    1. Int sinx dx = - cos x dx
    2. Int sin^3 x dx
    = Int (1-cos^2 x) sin x dx
    = Int sin x dx + Int cos^2 x (-sin x ) dx
    = - cos x + C + Int u^2 du
    = - cos x + C + 1/3 u^3 + C
    = - cos x + C + 1/3 cos^3 x +C
    = - cos x + 1/3 cos^3 x + C

    ReplyDelete
  24. Chyntia AvantikaMay 6, 2020 at 8:14 AM

    1. 1/2x - 1/4sin 2x + c
    2. -cos x + 1/3 cos^3x + c

    ReplyDelete
  25. MecaMay 6, 2020 at 8:34 AM

    Assalamualaikum warahmatullahi wabarakhatuh. Baiklah saya akan menjawab pertanyaan no 2
    Int sin³xdx=int sin²x sinx dx = int (1-cos²x) dcosx
    Misalkan cosx=u
    Int (1-u²)du=u- (1/3)u³ +c
    = Cosx -1/3 cos³x + c

    ReplyDelete
  26. Beti SeptianaMay 6, 2020 at 8:39 AM

    Beberapa identitas trigonometri yang diperluhkan adalah sebagai berikut:
    1.identitas pitagoras
    (Sin^2)x+(cos^2)x=1
    1+(tan^2)x=(sec^2)x
    1+(cot^2)x=(csc^2)x
    2.identitas setengah sudut
    (Sin^2)x=1-cos 2x/2
    (Cos^2)x=1+cos2x/2

    ReplyDelete
  27. Novi Siska SafitriMay 6, 2020 at 8:39 AM

    2. ∫ sin^3 x dx = ∫ sin^2 x (sinx) dx
    = ∫ (1-cos^2 x) (sinx) dx
    = ∫ sin x - cos^2 (sinx) dx
    = ∫sin x dx -∫cos^2 x sinx dx
    = - cos x + 1/3 cos^3 x + C

    ReplyDelete
  28. Rahmah IsnainiMay 6, 2020 at 8:43 AM

    1. Sin^2(x) dx
    = Integral (1-cos(2x)/2 dx
    =Integral 1/2 dx - integral cos(2x) dx
    = 1/2 x - sin(2x)/2 + C

    4. Integral cos(x) cos(4x)
    = Integral 1/2 (cos(-3x) + cos (5x)) dx
    =1/2 Integral cos(3x) + cos (5x) dx
    = 1/2 Integral cos(3x) dx + integral cos(5x) dx
    =1/2 (sin (3x)/3 + sin(5x)/5)
    =Sin(3x)/6 + sin (5x)/10 + C

    ReplyDelete
  29. deksu.12May 6, 2020 at 8:51 AM

    Dalam materi integrasi trigonometri ini, membahas tentang lima tipe integrasi trigobometri dan ditambah identitas trigonometri serta contoh-contoh dari setiap tipe yang sudah di jelaskan

    ReplyDelete
  30. Nike Apri YaniMay 6, 2020 at 8:54 AM

    Rumus integral trigonometri melibatkan fungsi trigonometri yg meliputi sin, cos, tan, secan, cosec, dan cotan. Integral fungsi trigonometri dapat diselesaikan dengan integral subtitusi maupun integral parsial, hanya saja untuk menentukan nilai integral trigonometri terdapat rumus baku hasil integral trigonometri.

    ReplyDelete
  31. MirandaMay 6, 2020 at 8:54 AM

    2.Int sin^3 x dx
    = Int sin ^2 x.sin x dx
    = Int (1-cos^2 x) d cos x
    = Cos x – 1/3 cos ^3 x +c

    ReplyDelete
  32. Vera anggelia oktaviaMay 6, 2020 at 9:04 AM

    Dalam melakukan integritas Trigonometri perlu teknik yang cepat dan terdapat beberapa identitas trigonometri yaitu: Identitas Pythagorean, Identitas Setengah sudut dan Tipe Tipe Integrasi Trigonometri
    Saya akan mencoba menjawab soal no 1 dan 2.
    1) =1/2 x-1/4 sin 2x+c
    2) = -cos x + 1/3 cos^3 x +c

    ReplyDelete
  33. Nehemia Ruth Octavia Silvia Sianturi May 6, 2020 at 9:19 AM

    Jawaban soal No. 1
    1. ∫ sin^2 x dx
    = ∫ 1 - cos^2 x / (2) dx
    = ∫ 1/2 dx - ∫ 1/2 cos 2x dx
    = 1/2 x - 1/4 sin 2x + C

    ReplyDelete
  34. Yemina BasariaMay 6, 2020 at 9:23 AM

    Dalam materi Integrasi Trigonometri, terbagi sub-bahasan yaitu : bentuk-bentuk integral trigonometri, identitas Pythagorean, identitas setengah sudut dan tipe-tipe integrasi trigonometri.
    Menjawab Soal :
    1) Integral sin^2 x dx
    = integral (1-cos(2x))/2 dx
    = integral 1/2 dx - integral 1/2 cos 2x dx
    = 1/2 x - 1/4 sin2x + C

    ReplyDelete
  35. Vella Sagita PutriMay 6, 2020 at 9:55 AM

    Tipe-tipe Integrasi Trigonometri
    1. ∫ sinn x dx dan ∫ cosn x dx
    2. ∫ sinm x cosn x dx
    3. ∫ sin mx cos nx dx ; ∫ sin mx sin nx dx ; dan ∫ cos mx cos nx dx
    4. ∫ tann x dx dan ∫ cotn x dx
    5. ∫ tanm x secn x dx dan ∫ cotm x cscn x dx

    Jawaban latihan soal :
    1. (1/2 )x - (1/4) sin2x +C
    2. Cos x - (1/3) (cos^3) x +C
    3. Cos^7 (3x) /21 - cos^5 (3x) /15 +C
    4. (1/10) sin5x + (1/6) sin3x +C
    5. (1/3) (tan^3)x - tan x + x +C
    6. -2 tan^-1/2 x + 2/3 tan^3/2 +C
    7. 2/3 (sec^3/2)x + 2(sec^-1/2)x +C

    ReplyDelete
  36. citra dewi romauli silabanMay 6, 2020 at 10:15 AM

    1.)∫sin^2 x dx
    sin (x)^2 = (1-cos(2x))/2
    ∫sin^2 x dx
    =∫(1-cos (2x))/2 dx
    = 1/2 × ∫ 1 – cos (2x) dx
    = 1/2 × (∫ 1 dx - ∫cos⁡(2x) dx)
    = 1/2 × (x - ∫cos⁡(2x) dx)
    = 1/2 × (x - sin⁡(2x)/2) + C
    = 1/2x - sin⁡(2x)/4 + C

    ReplyDelete
  37. Bunga Zuhri Al Fidiyah MMay 6, 2020 at 10:30 AM

    Jawaban nomor 5 :
    5. (1/3) (tan^3) x - tan x + x + C

    ReplyDelete
  38. Riska EndriMay 6, 2020 at 10:30 AM

    Saya akan menjawab no 1 dan 5
    1. ∫sin² x dx = In ½ (1-cos 2x) dx
    =½ In (1-cos 2x) dx
    =½ (x-½ sini 2x) + c
    =½x - ¼ sini 2x + c

    5. In tan⁴ x dx
    =tan³x/3 - In tan² x dx
    =tan³x/3 - In (sec² x-1) dx
    =tan³x/3 - (tan x-x) + c
    =x-1 1/3 tan³ x - tan x+c

    ReplyDelete
  39. Delviany UtariMay 6, 2020 at 10:31 AM

    Pembahasan rumus integral trigonometri melibatkan fungsi trigonometri yang meliputi sin, cos, tan, sec, cosec, dan cotan. Seperti cara menentukan nilai integral fungsi lainnya, integral fungsi trigonometri dapat juga diselesaikan dengan metode integral substitusi atau integral parsial.

    ReplyDelete
  40. Esti MarselaMay 6, 2020 at 10:35 AM

    Saya akan menjawab pertanyaan nomer 4. ∫cosx cos4x dx
    = ∫½[cos(4+1)x cos(4-1)x] dx
    = ∫½[cos5x cos3x] dx
    = ½∫cos5x dx + ½∫cos3x dx
    = ½(1/5)cos5x + ½(⅓)cos3x + C
    = 1/10 cos5x + 1/6 cos3x + C

    ReplyDelete
  41. Fajri RamadanMay 6, 2020 at 11:03 AM

    1.)∫sin^2 x dx
    =∫(1-cos (2x))/2 dx
    = 1/2 . ∫ 1 – cos (2x) dx
    = 1/2 . (∫ 1 dx - ∫cos⁡(2x) dx)
    = 1/2 . (x - ∫cos⁡(2x) dx)
    = 1/2 . (x - sin⁡(2x)/2) + C
    = 1/2x - 1/4sin⁡(2x) + C

    ReplyDelete
  42. Lola AzhariMay 6, 2020 at 11:41 AM

    Jawaban no 4.
    Integral dari cos x cos 4x dx = 1/2 integral [cos (1+5)x + cos (1-4)x ] dx
    = 1/2 integral [ cos 5x + cos (-3)x ] dx
    = 1/2 (1/5) sin 5x + (1/3) sin (3)x + C
    = 1/10 sin 5x + 1/6 sin 3x + C

    ReplyDelete
  43. Rahmat NursalimMay 6, 2020 at 5:45 PM

    Link kuis

    https://docs.google.com/forms/d/e/1FAIpQLSdcMpO0SKTM3XwdzQTHVcL4zR0aRv9y0iis9lOPwsk6f9qUnQ/viewform

    ReplyDelete
  44. Shelli claresiMay 6, 2020 at 6:22 PM

    Maaf pak baru komen skarang hp saya td dibenerin pak. Maaf pak 🙏🏻
    1. (1/2 )x - (1/4) sin2x +C
    2. Cos x - (1/3) (cos^3) x +C
    3. Cos^7 (3x) /21 - cos^5 (3x) /15 +C
    4. (1/10) sin5x + (1/6) sin3x +C
    5. (1/3) (tan^3)x - tan x + x +C

    ReplyDelete

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