Editorial - Matlan - Latihan Integral 2

Problem 1

(∫_^)(21*(7*x-9)^11*d(x))

u=7*x-9

d(u)/d(x)=7

d(x)=d(u)/7

(∫_^)(21*u^11⋅d(u)/7)=1/4*u^12+C=1/4*(7*x-9)^12+C

Problem 2

(∫_^)(25*x*(5*x-3)^4*d(x))

u=5*x-3

x=(u+3)/5

d(u)/d(x)=5

d(x)=d(u)/5

(∫_^)(25⋅(u+3)/5⋅u^4⋅d(u)/5)=(∫_^)((u^5+3*u^4)*d(u))

1/6*u^6+3/5*u^5=1/6*(5*x-3)^6+3/5*(5*x-3)^5+C

Problem 3

(∫_^)(x√(,16-x^2)*d(x))

u=16-x^2

x^2=16-u

d(u)/d(x)=-2*x

x*d(x)=d(u)/(-2)

(∫_^)(√(,u)⋅d(u)/(-2))=-1/2⋅1/3/2*u^(3/2)+C=(-u^3/2)/3+C=-1/3*(16-x^2)^(3/2)+C

Problem 4

(∫_^)(8*x√(3,2*x^2+5)*d(x))

u=2*x^2+5

d(u)/d(x)=4*x

x*d(x)=d(u)/4

(∫_^)(8⋅u^(1/3)⋅d(u)/4)=2⋅3/4⋅u^(4/3)+C=3/2*(2*x^2+5)^(4/3)+C

Problem 5

(∫_^)((6*x^2)/√(,x^3-27)*d(x))

u=x^3-27

d(u)/d(x)=3*x^2

x^2*d(x)=d(u)/3

(∫_^)(6/(u^(1/2))⋅d(u)/3)=2⋅2⋅u^(1/2)+C=4⋅(x^3-27)^(1/2)+C

Problem 6

(∫_^)(x/(x^2+1)*d(x))

u=x^2+1

d(u)/d(x)=2*x

x*d(x)=d(u)/2

(∫_^)(u^(-1)⋅1/2⋅d(u))

1/2*(∫_^)(1/u*d(u))

1/2*ln(u)+C=1/2*ln|x^2+1|+C

Problem 7

(∫_^)((8*x-4)/√(3,(x-3)*(x+2))*d(x))

(x-3)*(x+2)=x^2-x-6

u=x^2-x-6

d(u)/d(x)=2*x-1

d(u)=(2*x-1)*d(x)

(∫_^)((4*(2*x-1))/√(3,x^2-x-6)*d(x))=(∫_^)(4*u^(-1/3)*d(u))=6*u^2/3+C=6*(x^2-x-6)^(2/3)+C

Problem 8

(∫_^)(((9*x)*(x-10))/√(4,x^3-15*x^2-16)*d(x))

u=x^3-15*x^2-16

d(u)/d(x)=3*x^2-30*x=3*x*(x-10)

d(u)=3*x*(x-10)*d(x)

(∫_^)(3⋅u^(-1/4)*d(u))

3⋅4/3⋅u^(3/4)+C=4*(x^3-15*x^2-16)^(3/4)+C

Problem 9

(∫_^)(9/2*x^2√(,x^3+1)*d(x))

u=x^3+1

d(u)/d(x)=3*x^2

d(u)=3*x^2*d(x)

(∫_^)(3/2*u^(1/2)*d(u))

3/2⋅2/3⋅u^(3/2)+C=(x^3+1)^(3/2)+C

Problem 10

ƒ(x)=(∫_^)((6*x+3)*(x^2+x)^2*d(x))

ƒ(1)=8

d(u)/d(x)=2*x+1

ƒ(x)=(∫_^)(3*u^2)=(x^2+x)^3+C

ƒ(1)=8+C=8,∴C=0

ƒ(x)=(x^2+x)^3