Integral Parsial

Integral Parsial

Andaikan  U= ( X ) dan V = ( x )

Maka Dx [ u (x ).v (x)] =  u (x).v¹(x) + v (x).u¹(x)

ʃ Dx [u(x).v(x)] =ʃ u(x).v¹(x) + ʃ v(x).u¹ (x)

u(x).v(x)= ʃ u(x). v¹(x) + v(x).u¹(x)

ʃ u(x).v¹(x) = u(x).v(x) - ʃ v(x).u¹(x)

dv = v¹(x)        u = diturunkan

du = u¹(x)        v = diintegralkan

ʃ u(x) dv = u(x). v(x) - ʃ v(x)du

ʃ u dv = u.v - ʃ v du

contoh :

1.      ʃ x/v sin x/ dv dx

penyelesaian:

u = x

·         du/ dx = 1

·         Du = dx

Dv = sin x dx

Dv/dx= sin x

V = ʃ sin x dx

V= - cos x

 = ʃ u dv = u.v - ʃ v du

= ʃ u dv = x (- cos x ) - ʃ ( - cos x ) dx

= - x cos x + sin x + c

2.      ʃ x cos x dx

penyelesaian:

u = x

du = dx

dv= cos x dx

v= sin x

ʃ u dv = u.v - ʃ v du

           = x sin x - ʃ sin x dx

           = x sin x + cos x + c

Pengintegralan parsial untuk integral tentu :

ʃ^b_a  u dv = u.v ]^b_a - ʃ^b_a v du

contoh :

1.      ʃ₁^e In x dx

penyelesaian:

u = In x

du/dx = 1/x

du = 1/x dx

dv = dx

v = x

ʃ^e₁ In x dx = x In x │^e₁ - ʃ^e₁ x 1/x dx

              = [ ( e In e) – ( 1 In 1 )] - ʃ^e₁ dx

              =[ e In e – (1 In 1 ) ] -  [x│^e₁]

              = [ e In e – e ] – [ 1 In 1- 1]

              = [e (1) – e ] – [ 1.0.-1]

              = 0- (-1) = 1

Integral parsial berulang

Contoh

 ʃ x²/0  sin x/dx dx

u = x² du/dx= 2x

du = 2x dx

dx = sin x

v = - cos x

ʃ x² sin x = x² 9 – cos x) - ʃ - cos x. 2x dx

            = - x² cos x + 2 ʃ cos x xdx

            = - x² cos x + 2ʃ x cos x dx

            = - x² cos x + 2 ( x sin x + cos x)

            = - x² cos x + 2 x sin x + 2 cos x + c

Integral Trigonometri

ʃ cos (ax + b) = 1/a sin (ax +b)

ʃ cos ( 5x+3) = 1/5 sin ( 5x +3)

Ada 5 bentuk trignometri:

1.      ʃ sinⁿ x dx dan ʃ cosⁿ x dx

2.      ʃ sinⁿ x cosⁿ xdx

3.      ʃ tanⁿ x dx dan ʃ cotⁿ xdx

4.      ʃ tan^mx secⁿ xdx dan ʃ cot^m x cscⁿ x dx

5.      ʃ sin mx cos nx dx, ʃ sin mx sin nx dx dan ʃ cos mx cos nx dx

contoh:

1.      ʃ sin⁵ x dx

= ʃ sin⁴ x. sin x dx

= ʃ (sin²x)².  Sin x dx

= ʃ ( 1- cos² x)² sin x dx

= ʃ (1-2 cos² x + cos⁴ x) sin xdx

= - ʃ ( 1-2 cos² x+ cos⁴x) d ( cos x)

= -[ ʃ d ( cos x ) -ʃ 2 cos²x d (cos x) +nʃ cos⁴ x d(cos x)

= - [ cos x – 2 (1/3 cos ³x)+ 1/5 cos ⁵x

= - cos x +2/3 cos³x – 1/5 cos⁵ x+c

bentuk 2 :

ʃ  sin^m x cosⁿ x dx

a.       m atau n ganjil

= ʃ  sin³ x cos^-4 x dx

= ʃ  sin² x sinx . cos^-4  x dx

= ʃ ( 1- cos² x ). Cos^{. -4}  x sin x dx

= -ʃ ( cos^-4 x- cos^-2 x )d  ( cos x)

= - [ ʃ ( cos^-4 x d  ( cos x) - ʃ ( cos^-2 x d  ( cos x) -]

            = - [ - 1/3 cos^-3 x – ( - cos^-1 x )]

            =1/3 cos^-3   x – cos^-1 x+ c

            = 1/3 ( 1/ cos3^x) – ( 1/ cos x) +c

            = 1/3 sec³x- sec x+c  

Bentuk yang ketiga

ʃ tanⁿ x dx  dan ʃ cotⁿ x dx 

tan² x = sec² x - 1

cot² x = csc² x - 1

contoh:

ʃ cot⁴ x dx  = ʃ  (cot²  x . cot²  x ) dx

                 = ʃ  cot²  x ( csc²  x – 1 ) dx

                 = ʃ  (cot²  x . csc²  x- cot² x ) dx

                 =  ʃ  (cot²  x . csc²  x) -  ʃ  cot² x dx

                 =  ʃ  (cot²  x . csc²  x -  ʃ  (csc² x - 1) dx

                 =  ʃ  cot²  x d ( - cot  x ) -  ʃ ( csc² x – 1 ) dx

                 = - ʃ  cot²  x d ( cot  x ) -  ʃ csc² x dx -  ʃ dx

                 = - 1/3 cot³x – ( - cot x) – x+c

                 = - 1/3 cot³x + cot x – x+ c

Bentuk yang keempat

ʃ tan^m x secⁿ x dx,  ʃ cot^m x cscⁿ x dx

 

tan² x = sec² x - 1

sec² x = tan² x +1

cot² x = csc² x - 1

Contoh

a.       m sembarang dan n genap

ʃ  tan^-3/2  x . csc⁴  x  dx

     = ʃ  tan^-3/2  x ( sec² x) sec² x )dx

     = ʃ  tan^-3/2  x ( tan² x + 1) sec² x dx

     = ʃ (( tan^1/2  x  sec² x )  + tan^-3/2 x  sec² x ))

     = ʃ  tan^1/2  x d ( tan x)  + ʃ tan^-3/2 x d (tan x)

     = 2/3 tan^3/2 x + (-2 tan ^-1/2  x) + c

      = 2/3 tan^3/2 x – 2 tan^-1/2   + c