Integral Substitusi Yang Merasionalkan

INTEGRAL TRIGONOMETRI SELAIN SIN DAN COS

gimana sih menyelesaikan integral trigonometri selain sin dan cos?

Catatan : 1 + tan² x = sec² x

            cot² x + 1 = cosec² x

contoh :1) ʃ tan⁶ x dx

            =ʃ tan⁴ x. tan² x dx

            = ʃ tan⁴ x (sec² x – 1) dx

            = ʃ tan⁴  x. sec² x – tan⁴ x dx

            = ʃ tan⁴ x. sec² x- tan² x (sec² – 1) dx

            = ʃ tan⁴ x . sec² x – tan² x. sec² x + tan² x dx

            = ʃ tan⁴ x. sec² x – tan² x. sec² x + sec² x – 1 dx

                        Substitusi                                aturan biasa

            Missal : u = tan x

                        du = sec² x dx

            = ʃ tan⁴ x sec² x dx + ʃ tan² x sec² x dx + ʃ sec²  x -1 dx

            = ʃ u⁴ du - ʃ u² du + tan x – x + C

            = ʃ 1/5 tan⁵ x - 1/3 tan³ x + tan x – x + C

            2) ʃ cot³ x dx 

            = ʃcot x. cot²  x dx

            = ʃ cot x ( cosec² x -1) dx

            = ʃ cot x . cosec² x dx – ʃ cot x dx

            = ʃ cot x cosec² x dx - ʃ cos x / sin x dx

            Missal : u = cot x

                        du = cos² x dx

            missal : u = sin x

                        du = cos x dx

            = ʃ -1/2 cot² x – | ln sin x| + C

-          ʃ tan^m x . secⁿ x dx

-          ʃ cot^m x . cosecⁿ x dx

1)      n bilangan genap positif , m sembarang bilangan

contoh :

ʃ cot ^-1/2 x . cosec⁴ x dx

=ʃ cot ^-1/2 x . (1 + cot² x) . cosec² x dx

= ʃ (cot ^-1/2 x + cot ^3/2 x ) . cosec² x dx

      Missal : u = cot x

                  du = - cosec² x dx

= - ʃ u ^-1/2 + u ^3/2 du

= - 2 cot ^1/2 x – 2/5 cot ^3/2 x + C

2)      m bilangan ganjil positif , n sembarang bilangan

contoh :

ʃ tan³ x . sec x dx

= ʃ tan² x. tan x.sec x. sec x dx

= ʃ (sec² x + 1). . tan x. sec x dx

= ʃ sec² x . tan x. sec x dx + ʃ tan x. sec x dx

= ʃ sec² x . tan x. sec x dx + ʃ sin x / cos² x dx

      Missal : u = sec x

                  du = tan x . sec x dx

      missal : u = cos x

                  du = - sin x dx

= ʃ u² du + ʃ  - du/u²

= 1/3 u³ + C + 1/u + C

= 1/3 sec³ x + 1/cos x + C

selamat berlatih :))

INTEGRAL TRIGONOMETRI ( PANGKAT GANJIL, GENAP, DAN LANJUTAN )

 nah, sakarang kita bahas integral trigonometri yang sederhana dulu, seperti :

-          PANGKAT GENAP

Contoh :

ʃ sin ² x dx = ʃ ½ (1-cos 2x) dx                         

                   = ½ ʃ 1- cos 2x dx  
                                                          

                  = ½ (x-1/2 sin 2x) + C   
                                 

cos 2x   = cos ² x – sin ² x

   = 1- 2 sin ² x

       2 sin² x  = 1- cos 2x 
          Sin² x = ½ (1-cos 2x)
                                                                         

-          PANGKAT GANJIL

Contoh :

ʃcos⁵ x dx = ʃ cos⁴ x . cos x dx  
                        

                 = ʃ ( 1 - sin² x)² . cos x dx     
                             

                  = ʃ (1 - 2sin² x + sin⁴ x ). Cos x dx     
             

            Missal: u=sin x                                                 

                        du = cos x dx

                  = ʃ 1 – 2u² + u⁴ du  

                                                       

                  = u – 2/3 u + 1/5 u⁵ + C

                  = sin x – 2/3 sin³ x + 1/5 sin⁵ x + C
cos² x + sin² x = 1
    cos² x = 1- sin² x
   (cos² x)² = (1- sin² x)²
  cos⁴ x = (1- sin² x)²

 

-          PANGKAT LANJUTAN

Contoh :

ʃ Sin⁵ x . cos^-3 x dx  (syarat salah satu pangkat harus ganjil positif)

= ʃsin⁴ x. cos^-3 x. sin x dx

= ʃ (1 – cos² x )² . cos^-3x. sin x dx

= ʃ (cos^-3 x – 2 cos ^-1 x + cos x ) . sin x dx

Missal :         u = cos x

                        du = -sin x dx

= ʃ u^-3 – 2 u^-1 + u du

= - ʃ (-1/2 u^-2 – 2 |ln u| + ½ u²) + C

= ½ cos^-2 x + 2 |ln cos x| - ½ cos² x + C

Contoh lain :

ʃsin² x. cos² x dx (kedua pangkat harus genap positif) 

= ʃ1/2 (1- cos 2x ) . ½ (1+cos 2x) dx

=1/4 ʃ 1- cos² 2x dx

=¼ ʃ ½ (1 – cos 4x) dx

=1/8 ʃ 1 – cos 4x dx

=1/8 (x – ¼ sin 4x ) + C

selamat berlatih :))