Tentukanlah integral dari a. ∫[e^(3/x)/x^2] dx b. ∫x^2/(x^3+2) dx

Tentukanlah

a. ∫[e^(3/x)/x^2] dx
b. ∫x^2/(x^3+2) dx

Jawab:

a. Misalkan:
u = 3/x
dx = (-du x^2)/3

maka:

∫[e^(3/x)]/[x^2] dx= ∫e^u/x^2 [((-du x^2)/3)]

=-1/3 ∫e^u du

=-1/3 (e^u )

=-1/3 e^(3/x)+C

b. Misalkan:

u = x^3+2
dx = du/(3x^2 )

maka:

∫x^2/(x^3+2) dx =∫x^2/u (du/(3x^2 ))

=1/3 ∫du/u

=1/3 (ln u)+C

=1/3 ln⁡(x^3+2)+C

Tentukan turunan pertama dari  x^2 sin ⁡(xy) + y = x

Tentukan turunan pertama dari  x^2  sin⁡(xy)+ y=x

Jawab:

d/dx(x^2  sin(xy)+d/dx (y) = d/dx (x)

Pertama Turunkan: d/dx(sin⁡(xy)

d/dx(sin⁡(xy)=[ (1)(y)+ (x)(dy/dx)][cos⁡(xy)]

=(y+x dy/dx)(cos⁡(xy)

=y cos⁡(xy)+x dy/dx  cos(⁡xy)

Maka:

d/dx(x^2 sin(xy))+ d/dx (y)= d/dx (x)

2x(sin(xy))+x^2 (y cos⁡(xy)+ x dy/dx cos⁡(xy)+dy/dx =1

2x sin(xy)+x^2 ycos(xy)+x^3 dy/dx cos⁡(xy)+dy/dx = 1

dy/dx [x^3  cos⁡(xy)+1]= 1-2xsin(xy) – x^2 ycos(xy)

dy/dx = [1-2xsin(xy)-x^2 ycos(xy)]/[x^3  cos⁡(xy)+1]

Tentukan turunan pertama dari sin⁡(xy )+x^2=y^2+ 1

Tentukan turunan pertama dari sin⁡(xy )+x^2=y^2+ 1

Jawab:

d/dx(sin(xy)+d/dx (x^2 )= d/dx (y^2 )+d/dx(1)

Pertama Turunkan: d/dx(sin⁡(xy)

d/dx(sin⁡(xy)=[ (1)(y)+ (x)(dy/dx)][cos⁡xy]

=(y+x dy/dx)(cos⁡(xy)

=y cos⁡(xy)+x dy/dx cos⁡(xy)

Maka:

d/dx(sin(xy)+ d/dx (x^2 )= d/dx (y^2 )+d/dx(1)

y cos⁡(xy)+x dy/dx cos⁡xy + 2x = 2y dy/dx + 0

x dy/dx cos⁡(xy)-2y dy/dx = -y cos⁡(xy)-2x

dy/dx (x cos⁡(xy)-2y)= -y cos⁡(xy)-2x

dy/dx = (-y cos⁡(xy)-2x)/(x cos⁡(xy)-2y)

Tentukan turunan pertama dari tan (xy) – 2y = 0

Tentukan turunan pertama dari tan (xy) – 2y = 0

Jawab:

d/dx(tan(xy)-d/dx(2y)=d/dx(0)

Pertama Turunkan: d/dx(tan⁡(xy)

d/dx(tan⁡(xy)=[(1)(y)+ (x)(dy/dx)][sec^2⁡(xy)]

=(y+x dy/dx)(sec^2⁡ (xy)

=y sec^2⁡(xy) + x dy/dx sec^2⁡(xy)

Maka:
d/dx(tan(xy)-d/dx(2y)= d/dx()

y sec^2⁡(xy) + x dy/dx sec^2(⁡xy) -2 dy/dx = 0

x dy/dx sec^2(⁡xy)-2 dy/dx = -y sec^2⁡xy

dy/dx(x sec^2⁡(xy)-2) = -y sec^2⁡(xy)

dy/dx = -y sec^2⁡(xy))/(x sec^2⁡(xy)-2)

Turunan pertama dari f(x) = (1 + 1/x^2 )(1+1/x^3 )

Turunan pertama dari f(x) = (1 + 1/x^2 )(1+1/x^3 )

Jawab:

f(x) = U*V

f^’ (x)= U^’ V* UV^’

Maka:

f(x) = (1 + 1/x^2 )(1+1/x^3 )

f^’ (x)=((0-2x)/x^4 )((0-3x)/x^6 )

f^’ (x)=(2x/x^4 )(3x/x^6 )

f^’ (x)=((6x^2)/x^10 )

f^’ (x)=(6/x^8 )