Untuk integral dari invers fungsi trigonometri, saya akan memanfaatkan Integral Parsial, Integral Sustitusi dan sifat dasar dari Integral Sustitusi Trigonometri. Integral yang dibahas dalam tulisan ini adalah integral dari arc sin x, arc cos x, arc tan x, arc cosec x, arc sec x dan arc cotan x. Berikut integral dari fungsi tersebut.
arc sin x dx = …
ambil : u = arc sin x du = dx [bukti]
dv = dx v = x
arc sin x dx = x arc sin x – x dx
misal : a = 1 – x2 da = -2x dx
= x arc sin x – x
= x arc sin x + a-1/2 da
= x arc sin x + a1/2 + C
= x arc sin x + + C
arc sin x dx = x arc sin x + + C
.
arc cos x dx = …
ambil : u = arc cos x du = dx [bukti]
dv = dx v = x
arc cos x dx = x arc cos x – x dx
misal : a = 1 – x2 da = -2x dx
= x arc cos x – x
= x arc cos x – a-1/2 da
= x arc cos x – a1/2 + C
= x arc cos x – + C
arc cos x dx = x arc cos x – + C
.
arc tan x dx = …
ambil : u = arc tan x du = dx [bukti]
dv = dx v = x
arc tan x dx = x arc tan x – x dx
misal : a = 1 + x2 da = 2x dx
= x arc tan x – x
= x arc tan x – da
= x arc tan x – ln|a| + C
= x arc tan x – ln|1 + x2| + C
arc tan x dx = x arc tan x – ln|1 + x2| + C
.
arc csc x dx = …
ambil : u = arc csc x du = dx [bukti]
dv = dx v = x
arc csc x dx = x arc csc x – x dx
= x arc csc x + dx
misal : x = sec t dx = sec t tan t dt
= x arc csc x + sec t tan t dt
= x arc csc x + sec t tan t dt
= x arc csc x + sec t dt
= x arc sec x + ln|sec t + tan t| + C [bukti]
arc csc x dx = x arc sec x + ln|sec t + tan t| + C
.
arc sec x dx = …
ambil : u = arc sec x du = dx [bukti]
dv = dx v = x
arc sec x dx = x arc sec x – x dx
= x arc sec x – dx
misal : x = sec t dx = sec t tan t dt
= x arc sec x – sec t tan t dt
= x arc sec x – sec t tan t dt
= x arc sec x – sec t dt
= x arc sec x – ln|sec t + tan t| + C [bukti]
.
arc sec x dx = x arc sec x – ln|sec t + tan t| + C
.
arc cot x dx = …
ambil : u = arc cot x du = dx [bukti]
dv = dx v = x
arc cot x dx = x arc cot x – x dx
misal : a = 1 + x2 da = 2x dx
= x arc cot x + x
= x arc cot x + da
= x arc cot x + ln|a| + C
= x arc cot x + ln|1 + x2| + C
arc cot x dx = x arc cot x + ln|1 + x2| + C
Gak ngerti -__-
Thanks artikelx… http://matematikakubisa.blogspot.com
maaf bang, hasil integral arc tan x nya kurang 1/2 di ln(x kuadrat + 1)
iya, makasi koreksinya mas 🙂
eh ternyata blognya aim.. udah baca2 baru ngeh setelah liat gambar penulisnya 😀
eh ada aini, makasi sudah berkunjung diblog yg masih jauh dari sempurna ini ai 🙂