Soal integral

yang pinter-pinter, entah hanya ngerasa ato emang pinter beneran, tolong ya!

soal:
Tunjukkan bahwa:
a.

b.

tolong nih

Bls: Soal integral

Pake tulisan biasa aja ya, mas...)
Aku rada gaptek kalo harus pake fitur penulisan yang sudah disediakan itu...

1. sin^4(x)

= [sin^2(x)]^2

= [(1 -cos(2x)/2)]^2

= 1/4[1 + cos^2(2x) - 2cos(2x)]

= 1/4[1 - 2cos(2x) + (1 + cos(4x)/2 ]

= (1/4) - (1/2)cos(2x) + (1/8) + (1/8)cos(4x)

= (3/8) - (1/2)cos(2x) + (1/8)cos(4x)

maka ∫sin^4(x)dx = 3/8∫dx - 1/2∫cos(2x) dx + 1/8∫cos(4x) dx

∫sin^4(x)dx = (3/8)x - (1/2)sin(2x)(1/2) + (1/8)sin(4x)(1/4) + C

∫sin^4(x)dx = (3/8)x - (1/4)sin(2x) + (1/32)sin(4x) + C



2. ∫ cos^4(x) dx = ∫ (cos^2x)^2 dx

karena cos^2x = (1/2)[1 + cos(2x)], maka menjadi

∫ {(1/2)[1 + cos(2x)]}^2 dx = ∫ (1/4)[1 + cos(2x)]^2 dx

∫ (1/4)[1 + 2cos(2x) + cos^2(2x)] dx

= (1/4) ∫ dx + (1/4) 2 ∫ cos(2x) + (1/4) ∫ cos^2(2x) dx

= (1/4) ∫ dx + (1/2) ∫ cos(2x) + (1/4) ∫ cos^2(2x) dx

= (1/4)x + (1/2) (1/2)sin(2x) + (1/4) ∫ cos^2(2x) dx

= (1/4)x + (1/4)sin(2x) + (1/4) ∫ cos^2(2x) dx

dari hasil di atas, maka

cos^2(2x) = (1/2){1 + cos[2(2x)]} = (1/2)[1 + cos(4x)]

oleh karena itu,

(1/4)x + (1/4)sin(2x) + (1/4) ∫ (1/2)[1 + cos(4x)] dx

= (1/4)x + (1/4)sin(2x) + (1/4)(1/2) ∫ [1 + cos(4x)] dx

= (1/4)x + (1/4)sin(2x) + (1/8) ∫ dx + (1/8) ∫ cos(4x) dx

= (1/4)x + (1/4)sin(2x) + (1/8)x + (1/8) (1/4)sin(4x) + C

= [(1 + 2)/8]x + (1/4)sin(2x) + (1/32)sin(4x) + C

Maka,

∫ cos^4(x) dx = (3/8)x + (1/4)sin(2x) + (1/32)sin(4x) + C


-dipi-

Bls: Soal integral

ih lihat jawabannya non dipi aja kau udah puyeng.....

Bls: Soal integral

@non dipi:
wohoho, TOP dah, tadi pagi untungnya aku sempet nanya temenku yg juara olim, dan hasilnya sama, repu deh bwt non!

Bls: Soal integral

Ampun God.......... puyeng liatnya... klasih reppu aja deh ke tante dipi dan den musthaf

Bls: Soal integral

Jadi kangen masa sma dl,

Krn guru Mat gw cinta bgt ma 'Integral' khususny 'Integral Parsial'..

Dohh, tiap hr pr Integral itu berlembar2 boo...
^^

Bls: Soal integral

pusing

wow mumet bener gan
<:||

waduh ga ngerti