Question

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Answer 1

Integ cos(phi - 2x) cot(2x) dx =
integ -cos2x .cot2x  dx ..................ingat rumus integ cosecxcotx = -cosecx + c
sehingga
-integ cos2xcot2x dx = - (-cosec 2x ) + c
= cosec 2x + c

Answer 2

Identitas :
cos(π - 2x) = -cos(2x)
cot = cos/sin
cos² = 1 - sin²

int cos(π - 2x) cot(2x) dx
= - int cos(2x) cos(2x)/sin(2x)  dx
= - int cos²(2x)/sin(2x)  dx
= - int (1 - sin²(2x))/sin(2x)  dx
= int (sin²(2x) - 1)/sin(2x)  dx
= int sin²(2x)/sin(2x) dx  - int 1/sin(2x)  dx
= int sin(2x)  -  int cosec(2x) dx

jelas, int sin(2x) = -1/2 cos(2x)

untuk case int cosec(2x) :

int cosec(2x) dx
= int cosec(2x) * (cosec(2x)+cot(2x))/(cosec(2x)+cot(2x)) dx
= int (cosec²(2x) + cosec(2x)cot(2x)) / (cosec(2x)+cot(2x)) dx
misal :
u = cosec(2x) + cot(2x)
du = (-2cosec(2x)cot(2x) - 2 cosec²(2x)) dx
-1/2 du = (cosec(2x)cot(2x) + cosec²(2x)) dx
so,
= -1/2 int 1/u  du
= -1/2 ln(u)
= -1/2 ln(cosec(2x) + cot(2x))

jadi,
int sin(2x) dx - int cosec(2x) dx
= -1/2 cos(2x) - (-1/2 ln(cosec(2x)+cot(2x))  + C
= -1/2 cos(2x) + 1/2 ln(cosec(2x)+cot(2x)) + C
= -1/2 cos(2x) + 1/2 ln( (1 + cos(2x))/sin(2x) ) + C
= -1/2 cos(2x) + 1/2 ln( 2sin²x / sin(2x)) + C
= -1/2 cos(2x) + 1/2 ln(2sin²x / 2sin(x)cos(x) ) + C
= -1/2 cos(2x) + 1/2 ln(sin(x)/cos(x)) + C
= -1/2 cos(2x) + 1/2 ln(sin(x)) - 1/2 ln(cos(x)) + C