Editorial - Matlan - Latihan Integral Trigonometri

Problem 1

(∫_^)(2*x+6*sin(x)*d(x))=x2-6*cos(x)+C

Problem 2

(∫_^)(4*x3+2*sec2(x)*d(x))=x4+2*tan(x)+C

Problem 3

(∫_^)(6*x2-2*cos(2*x)*d(x))=2*x3-(∫_^)(2*cos(2*x)*d(x))+C

u=2*x

d(u)/d(x)=2

d(x)=d(u)/2

2*x3-sin(2*x)+C

Problem 4

(∫_^)(10*x(3/2)-6*sin(3*x)*d(x))=4*x(5/2)+2*cos(3*x)+C

Problem 5

(∫_^)(6*sin(3*x)-2*sec(4*x)*tan(4*x)*d(x))=-2*cos(3*x)-1/2*sec(4*x)+C

Problem 6

(∫_^)(2*cos(2*x)+6*csc(3*x)*cot(3*x)*d(x))=sin(2*x)-2*csc(3*x)+C

Problem 7

(∫_^)(12*sin(4*x+2)-3*cos(3*x-1)*d(x))

-3*cos(4*x+2)-sin(3*x-1)+C

Problem 8

(∫_^)(sec(2*x+1)*tan(2*x+1)-csc(4*x-3)*d(x))

1/2*sec(2*x+1)-(∫_^)(csc(4*x-3)*d(x))

(∫_^)(csc(x)*d(x))=(∫_^)(csc(x)⋅(csc(x)+cot(x))/(csc(x)+cot(x))*d(x))

(∫_^)((csc2(x)+csc(x)*cot(x))/(csc(x)+cot(x))*d(x))

u=csc(x)+cot(x)

d(u)/d(x)=-csc(x)*cot(x)-csc2(x)=-(csc(x)*cot(x)+csc2(x)) (form ini sudah ditemukan)

(∫_^)(-1/u*d(u))=-ln|csc(x)+cot(x)|

Identitas:

(∫_^)(csc(x)*d(x))=-ln|csc(x)+cot(x)|

1/2*sec(2*x+1)+1/4*ln|csc(4*x-3)+cot(4*x-3)|+C

Problem 9

(∫_^)(tan(3*x+2)+csc(2*x-1)*cot(2*x-1)*d(x))

(∫_^)(tan(3*x+2)*d(x))-1/2+csc(2*x-1)*C

(∫_^)(tan(x)*d(x))=(∫_^)(sin(x)/cos(x)*d(x))

u=cos(x)

d(u)/d(x)=-sin(x)

(∫_^)(sin(x)/u⋅d(u)/(-sin(x)))=-(∫_^)(1/u*d(u))=-ln|u|+C

-1/3*ln|cos(3*x+2)|-1/2*csc(2*x-1)+C

Problem 10

ƒ(x)=?

ƒ(1/6*π)=4

(ƒ^′)(x)=2√(,3)*sec2(2*x+π/3)

ƒ(x)=(∫_^)(2√(,3)*sec2(2*x+π/3)*d(x))=√(,3)*tan(2*x+π/3)+C

ƒ(π/6)=√(,3)*tan(π/3+π/3)+C=√(,3)*tan((2*π)/3)+C

tan((2*π)/3)=-√(,3)

√(,3)⋅(-√(,3))+C=4

∴C=7

∴ƒ(x)=√(,3)*tan(2*x+π/3)+7

Problem 11

uh oh

(∫_^)(x*cos(x2)*d(x))

cos(x2) tidak ada integralnya maka x2=u

u=x2

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

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

(∫_^)(x*cos(u)⋅d(u)/(2*x))=(∫_^)(cos(u)/2*d(u))=1/2*sin(u)+C=1/2*sin(x2)+C

Problem 12

u=sin(x)

d(u)/d(x)=cos(x)

d(x)=d(u)/cos(x)

(∫_^)(u2*d(u))=1/3*sin3(x)+C

Problem 13

(∫_^)(cos(x)/sin2(x)*d(x))

u=sin(x)

d(u)/d(x)=cos(x)

d(x)=d(u)/cos(x)

(∫_^)(cos(x)/(u2)d(u)/cos(x))=(∫_^)(1/(u2)*d(u))=(∫_^)(u(-2)*d(u))=-u(-1)+C=-1/sin(x)+C

Problem 14

u=6*x3

d(u)/d(x)=18*x2

(∫_^)((2*x2*sin(u)*d(u))/(18*x2))=-1/9*cos(u)+C=-1/9*cos(6*x3)+C

Problem 15

u=2*x2

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

(∫_^)((x*sec2(u)*d(u))/(4*x))=1/4*tan(2*x2)+C

Problem 16

:)

u=1/x

d(u)/d(x)=-1/(x2)

d(x)=d(u)⋅-x2

(∫_^)(2/(x2)*sin(u)*d(u)⋅-x2)=-2*(∫_^)(sin(u)*d(u))=2*cos(u)=2*cos(1/x)+C

Problem 17

u=√(,x)

d(u)/d(x)=1/(2√(,x))

d(x)=d(u)⋅2√(,x)

(∫_^)(2/√(,x)*cos(u)*d(u)⋅2√(,x))=4*(∫_^)(cos(u)*d(u))=4*sin(√(,x))+C

Problem 18

(∫_^)(8*tan(4*x)*d(x))

u=cos(4*x)

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

(∫_^)(8⋅sin(4*x)/u⋅d(u)⋅1/(-4*sin(4*x)))=-2*(∫_^)(1/u*d(u))
=-2*ln|cos(4*x)|+C

Problem 19

(∫_^)(4*sec(2*x)*tan(2*x)-tan(2*x)*d(x))=2*sec(2*x)+1/2*ln|cos(2*x)|+C

Problem 20

(∫_^)(tan2(x)+2*sin(x)*cos(x)*d(x))

2*sin(x)*cos(x)=sin(2*x)

tan2(x)=sec2(x)-1

(∫_^)(sec2(x)-1+sin(2*x)*d(x))=tan(x)-x-cos(2*x)/2+C

Problem 21

12*sin(3*x)*cos(3*x)=6*sin(6*x)

(∫_^)(6*sin(6*x)*d(x))=-cos(6*x)+C

Problem 22

u=cos(x)

d(u)/d(x)=-sin(x)

d(u)=-d(x)*sin(x)

(∫_^)(12*cos5(x)*sin(x)*d(x))=(∫_^)(-12*u5*d(u))=-2*u6=-2*cos6(x)+C

Problem 23

sin(a)*cos(b)=1/2*[sin(a+b)+sin(a-b)]

sin(7*x)*cos(6*x)=1/2*(sin(13*x)+sin(x))

(∫_^)(4*sin(7*x)*cos(6*x)*d(x))=2*(∫_^)((sin(13*x)+sin(x))*d(x))
=-2/13*cos(13*x)-2*cos(x)+C

Problem 24

INTEGRATION BY PARTS.

(∫_^)((3*x+1)*cos(2*x)*d(x))=(3*x+1)*(1/2*sin(2*x))+(-1)*(3)*(-1/4*cos(2*x))
=(3*x*sin(2*x)+sin(2*x))/2+3/4*cos(2*x)+C

Problem 25

(∫_^)(x*cos(2*x-1)*d(x))

INTEGRATION BY PARTS.

(∫_^)(x*cos(2*x-1)*d(x))=x/2*sin(2*x-1)+(-1)⋅(-1/4*cos(2*x-1))+C
=x/2*sin(2*x-1)+1/4*cos(2*x-1)+C

Problem 26

(∫_^)((x+3)*cos(2*x-π)*d(x))

cos(2*x-π)=cos(-(π-2*x))=cos(π-2*x)=-cos(2*x)

-1*(∫_^)((x+3)*cos(2*x)*d(x))

INTEGRATION BY PARTS.

-1*(∫_^)((x+3)*cos(2*x)*d(x))=-((x+3)/2*sin(2*x)+(-1)⋅(-1/4*cos(2*x)))=-(x+3)/2*sin(2*x)-1/4*cos(2*x)+C

Problem 27

INTEGRATION BY PARTS.

(∫_^)((x2+1)*cos(x)*d(x))=(x2+1)*sin(x)+(-1)*(-2*x*cos(x))+2⋅(-sin(x))
=x2*sin(x)+2*x*cos(x)-sin(x)+C

Problem 28

(∫_^)(sin(x)/sin5(3*x)*d(x))

(SKIP)

Problem 29

u=sin(2*x)

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

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

Problem 30

u=sin(π*t)

d(u)=d(t)*π*cos(π*t)

(∫_^)(cos(π*t)*cos(sin(π*t))*d(t))=1/π*(∫_^)(cos(u)*d(u))
=1/π*sin(u)
=1/π*sin(sin(π*t))+C