Editorial - Matlan - Latihan Integral 1
Problem 1
(∫_^)(2*x+1)*d(x)=(2*x^2)/2+1/1*x+C=x^2+x+C
Problem 2
(∫_^)(x^2-x+3)*d(x)=1/3*x^3-(x^2)/2+3*x+C
Problem 3
(∫_^)(1/(x^4)+1/(x^(1/4)))*d(x)=(∫_^)(x^(-4)+x^(-1/4))*d(x)=-1/3*x^(-3)+4/3*x^(3/4)+C
Problem 4
(∫_^)(x^2*(x+2))*d(x)=(∫_^)(x^3+2*x^2)*d(x)=1/4*x^4+2/3*x^3+C
Problem 5
(∫_^)((x^2+3)^2)*d(x)=(∫_^)(x^4+6*x^2+9)*d(x)=1/5*x^5+2*x^3+9*x+C
Problem 6
(∫_^)((x^3-2*x)^2)*d(x)=(∫_^)(x^6-4*x^4+4*x^2)*d(x)=1/7*x^7-4/5*x^5+4/3*x^3+C
Problem 7
(∫_^)((x^5-x^2+1)/(x^2))*d(x)=(∫_^)(x^3-1+x^(-2))=1/4*x^4-x-x^(-1)+C
Problem 8
(∫_^)(10/(x√(5,x^2)))*d(x)=(∫_^)(10/(x⋅x^(2/5)))*d(x)=(∫_^)(10/(x^(7/5)))*d(x)=(∫_^)(10*x^(-7/5))*d(x)
(10*x^(-2/5))/(-2/5)+C=-5/2⋅10⋅x^(-2/5)+C=-25/(x^(2/5))+C
Problem 9
(∫_^)((6*x^2+4√(,x))/x)*d(x)=(∫_^)(6*x+4*x^(-1/2))*d(x)
3*x^2+8*x^(1/2)+C
Problem 10
(∫_^)((9√(3,x)+8√(9,x^2)+10*x)/√(3,x))*d(x)=(∫_^)(9+8*x^(-1/9)+10*x^(2/3))*d(x)
9*x+9*x^(8/9)+6*x^(5/3)+C
Problem 11
(∫_^)(((x-1)^2*(x+3))/(x√(,x)))*d(x)=(∫_^)((x^3+x^2-5*x+3)/(x^(3/2)))*d(x)
(∫_^)((x^(3/2)+x^(1/2)−5*x^(−1/2)+3*x^(−3/2)))*d(x)
2/5*x^(5/2)+2/3*x^(3/2)-10*x^(1/2)-6*x^(-1/2)+C
Problem 12
(∫_^)(((x^2-x^(-1))^2)/√(,x))*d(x)
(∫_^)((x^4-2*x+(x^)^(-2))/(x^(1/2)))*d(x)
(∫_^)(x^(7/2)-2*x^(1/2)+x^(-5/2))*d(x)
2/9*x^(9/2)-4/3*x^(3/2)-2/3*x^(-3/2)+C
Problem 13
(ƒ^′)(x)=12*(x+1)*(x+2)
ƒ(-3)=-18
ƒ(x)=?
Sol.
(ƒ^′)(x)=12*(x^2+3*x+2)
ƒ(x)=(∫_^)(12*(x^2+3*x+2))*d(x)=12*(1/3*x^3+3/2*x^2+2*x+C)
ƒ(-3)=12*(1/3⋅(-27)+3/2⋅(9)+2⋅(-3)+C)=-18+12*C
12*C=-18+18=0
∴C=0
ƒ(x)=12*(1/3*x^3+3/2*x^2+2*x+0)=4*x^3+18*x^2+24*x
Problem 14
y=ƒ(x)
(ƒ^′)(x)=3*x^2+2
ƒ(2)=5
ƒ(x)=?
Sol.
ƒ(x)=(∫_^)(3*x^2+2)*d(x)=x^3+2*x+C
ƒ(2)=8+4+C=5
C=5-8-4=-7
∴ƒ(x)=x^3+2*x-7
Problem 15
Gradien garis singgung di (x,y) adalah (2*x+3√(,x))
Kurva melalui (4,2)
Cari garis singgung di x=1
Sol.
ƒ(4)=2
(ƒ^′)(x)=2*x+3√(,x)
Find gradient of tangent at x=1
(ƒ^′)(1)=2+3=5
∴m=5
Find ƒ(x)
ƒ(x)=x^2+2*x^(3/2)+C
ƒ(4)=16+2⋅4^(3/2)+C=32+C=2
∴C=-30
Find ƒ(1)
ƒ(1)=1+2-30=-27
Tangent is at (1,-27)
y=m*x+b
y=5*x+b
-27=5+b
b=-32
∴y=5*x-32