Editorial - Matlan - Latihan Integral 1

Problem 1

(∫_^)(2*x+1)*d(x)=(2*x2)/2+1/1*x+C=x2+x+C

Problem 2

(∫_^)(x2-x+3)*d(x)=1/3*x3-(x2)/2+3*x+C

Problem 3

(∫_^)(1/(x4)+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

(∫_^)(x2*(x+2))*d(x)=(∫_^)(x3+2*x2)*d(x)=1/4*x4+2/3*x3+C

Problem 5

(∫_^)((x2+3)2)*d(x)=(∫_^)(x4+6*x2+9)*d(x)=1/5*x5+2*x3+9*x+C

Problem 6

(∫_^)((x3-2*x)2)*d(x)=(∫_^)(x6-4*x4+4*x2)*d(x)=1/7*x7-4/5*x5+4/3*x3+C

Problem 7

(∫_^)((x5-x2+1)/(x2))*d(x)=(∫_^)(x3-1+x(-2))=1/4*x4-x-x(-1)+C

Problem 8

(∫_^)(10/(x√(5,x2)))*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*x2+4√(,x))/x)*d(x)=(∫_^)(6*x+4*x(-1/2))*d(x)

3*x2+8*x(1/2)+C

Problem 10

(∫_^)((9√(3,x)+8√(9,x2)+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)=(∫_^)((x3+x2-5*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

(∫_^)(((x2-x(-1))2)/√(,x))*d(x)

(∫_^)((x4-2*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*(x2+3*x+2)

ƒ(x)=(∫_^)(12*(x2+3*x+2))*d(x)=12*(1/3*x3+3/2*x2+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*x3+3/2*x2+2*x+0)=4*x3+18*x2+24*x

Problem 14

y=ƒ(x)

(ƒ^′)(x)=3*x2+2

ƒ(2)=5

ƒ(x)=?

Sol.

ƒ(x)=(∫_^)(3*x2+2)*d(x)=x3+2*x+C

ƒ(2)=8+4+C=5

C=5-8-4=-7

∴ƒ(x)=x3+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)=x2+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