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Pengamiran ( Soalan Dan Jawapan )

 

1. Kamilkan  setiap yang berikut :

a)      ∫ x2 dx  = x2  / 3 + c

b)      ∫ x3 dx  = x4 / 4 + c

c)      ∫ (3x2 + 2x) dx = x3 + x2 + c

d)      ∫ ( 2x2 + 3x 1) dx = 2x3 /3 +3x2 / 2 x + c

e)      ∫ ( 4x3 x 2) dx = x4 x3 / 3 + c

f)        ∫ ( x + 1/x ) dx = ∫ x dx + ∫1/x dx = x2 / 2 + ln x + c

g)      ∫ ( 2/x ) dx = 2 ∫ 1/x dx = 2 ln x + c

h)      ∫ x2/3 dx = x5/3 / 5/3  + c = 3x5/3 /5 + c

i)        ∫ x2 ( x-1 ) dx = ∫ ( x3 x 2)  dx = x4 / 4 x3 / 3 + c

j)        ∫ ( 1 3x ) (1 x ) dx =  ∫ ( 1 4x + 3x2 ) dx = x 2x2 + x3 + c

k)       ∫ ( x3 + 1 ) / x2 dx = ∫ ( x + x -2) dx = x2 / 2 1 / x + c

l)        ∫ ( x 1 ) / x dx = ∫ ( 1 1/x ) dx = x ln x + c

2. Kamirkan bagi setiap fungsi tertentu

a)      Katakan a = sin x

Da / dx = kos x

Dx = da / kos x

∫ sin2 x . kos x dx

= ∫ a2 kos x  da / kos x

=  a3 / 3 + c

= sin3 / 3 + c

 

b)      Katakan x / 2 = a

Da / dx =

Dx = 2 da

∫ sin ( x/2 ) dx

= ∫ sin a 2 da

= 2 ∫ sin a da

= 2 ( - kos a ) + c

= - 2 kos ( x/2 ) + c

 

c)      Katakan a = 3x

Da / dx = 3

Dx = da / 3

∫ kos 3x dx

= ∫ kos a da/3

= 1/3 ∫ kos a da

= 1/3 ( sin a ) + c

= 1/3 sin 3x + c

 

d)      Katakan a = 2x 3

Da /dx = 2

Dx = da / 2

∫ dx / ( 2x 3 )

= ∫ da / 2 / a

= ∫ da /a

= ln a + c

= ln ( 2x + c ) + c

e)      Katakan a = x2 - 1  

Da /dx = 2x

Dx = da / 2x

∫ x / (  x2 - 1 ) dx

= ∫ x /a da/2x

= ∫ 1/a da

=   ln ( x2 - 1 ) + c

 

f)        Katakan a = x3 4

Da / dx = 3x2

Dx = da / 3x2

∫ 2x2 / ( x3 4 ) dx

= ∫ 2x2 / a a / 3x2

= 2/3 ∫ 1/a a

= 2/3 ln a + c

= 2/3 in ( x3 4 ) + c

 

g)      Katakan a = kos x

Da / dx = - sin x

Dx = da / - sin x

∫ tan x dx

= ∫ sin x / kos x  dx

= ∫ sin x / a .  da / - sin x

= -1 ∫ 1/a da

= -1 ln a + c

= -1 ln cos x + c

 

h)      Katakan a = 3x

Da / dx = 3

Dx = da / 3

∫ e3x dx

= ∫ ea da/3

= 1/3 ∫ ea da

= 1/3 ea  + c

            = e3x / 3 + c