gimana sih menyelesaikan integral trigonometri selain sin dan cos?
Catatan : 1 + tan2 x = sec2 x
cot2
x + 1 = cosec2 x
contoh :1) ʃ tan6 x dx
=ʃ
tan4 x. tan2 x dx
=
ʃ tan4 x (sec2 x – 1) dx
=
ʃ tan4 x. sec2 x –
tan4 x dx
=
ʃ tan4 x. sec2 x- tan2
x (sec2 – 1) dx
=
ʃ tan4 x . sec2 x – tan2 x. sec2 x +
tan2 x dx
=
ʃ tan4 x. sec2 x – tan2 x.
sec2 x + sec2 x – 1 dx
Substitusi aturan biasa
Missal : u = tan x
du = sec2 x dx
= ʃ tan4
x sec2 x dx + ʃ tan2 x
sec2 x dx + ʃ sec2 x -1 dx
= ʃ u4 du - ʃ u2
du + tan x – x + C
= ʃ 1/5 tan5 x - 1/3 tan3
x + tan x – x + C
2)
ʃ cot3 x dx
=
ʃcot x. cot2 x dx
=
ʃ cot x ( cosec2 x -1) dx
=
ʃ cot x . cosec2 x dx – ʃ cot x dx
= ʃ cot x
cosec2 x dx - ʃ cos x / sin x dx
Missal : u = cot x
du = cos2 x dx
missal : u = sin x
du = cos x dx
=
ʃ -1/2 cot2 x – | ln sin x| + C
-
ʃ tanm x . secn x
dx
-
ʃ cotm x . cosecn
x dx
1)
n bilangan genap positif , m sembarang bilangan
contoh :
ʃ cot -1/2 x . cosec4
x dx
=ʃ cot -1/2 x . (1 + cot2 x) . cosec2
x dx
= ʃ (cot -1/2 x + cot 3/2 x ) . cosec2
x dx
Missal : u = cot
x
du =
- cosec2 x dx
= - ʃ u -1/2 + u 3/2 du
= - 2 cot 1/2 x – 2/5 cot 3/2 x + C
2)
m bilangan ganjil positif , n sembarang bilangan
contoh :
ʃ tan3 x . sec x
dx
= ʃ tan2 x. tan x.sec x. sec x dx
= ʃ (sec2 x + 1). . tan x. sec x dx
= ʃ sec2 x . tan x. sec x
dx + ʃ tan x. sec x dx
= ʃ sec2 x . tan x. sec x dx + ʃ sin x / cos2 x
dx
Missal : u = sec x
du = tan x . sec x dx
missal
: u = cos x
du
= - sin x dx
= ʃ
u2 du + ʃ - du/u2
= 1/3 u3 + C + 1/u + C
= 1/3
sec3 x + 1/cos x + C
selamat berlatih :))
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