ma1201 matematika 2a - · pdf filema1101 matematika 1a author: asus created date: 1/30/2017...
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MA1201 MATEMATIKA 2A
Hendra GunawanSemester II, 2016/2017
18 Januari 2017
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Hendra Gunawan• Gedung CAS, Lt. 4, R. 3021
• Tel. 2502545 Pes. 112
• E-mail [email protected]
• Website http://personal.fmipa.itb.ac.id/hgunawan/
• Twitter @hgunawan82
1/22/2014 2(c) Hendra Gunawan
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Silabus MA1201
1. Teknik Pengintegralan
2. Bentuk Tak Tentu dan Integral Tak Wajar
3. Deret Tak Hingga
4. Irisan Kerucut, Persamaan ParametrikKurva, Geometri di Bidang & Ruang
5. Fungsi Dua & Tiga Peubah
6. Integral Lipat
7. Persamaan Diferensial Orde Dua1/22/2014 3(c) Hendra Gunawan
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Tujuan Umum Pembelajaran
Dengan mengikuti kuliah ini, mahasiswa diharapkanmemiliki:
1. Keterampilan teknis baku yang didukung oleh konsep, rumus, metode, dan penalaran yang sesuai;
2. Pola berpikir yang kritis, logis, dan sistematis, sertakreativitas dalam pemecahan masalah yang terkaitdengan matematika, khususnya kalkulus;
3. Kemampuan mengkomunikasikan hasil pemikiran danpekerjaannya baik secara lisan maupun tulisan;
4. Kesiapan untuk mempelajari matakuliah lain, yang memerlukan matematika sebagai prasyarat, secaramandiri.
1/22/2014 (c) Hendra Gunawan 4
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Hitunglah
hingga 5 angka desimal(di belakang koma).
Sebuah kotak terbuat daribahan dengan biaya Rp 25 per cm2 untuk bagian sisitegak dan tutupnya, danRp 40 per cm2 untuk bagianalasnya. Tentukan ukurankotak bervolume 2 m3 yang biayanya minimum.
Berapakah besar arus bilasaklar dihubungkan?
CONTOH PERMASALAHAN
1/22/2014 5(c) Hendra Gunawan
4.0
0
41 dxx
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Ujian, Kuis dan PR
• Ujian I (18 Maret) dan Ujian II (29 April), @45%
• PR/Tugas, Kuis, dan Keaktifan di Kelas, total 10%
Nilai Akhir dinyatakan dalam huruf:
35 ≤ D < 50 ≤ C < 57 ≤ BC < 65 ≤ B < 73 ≤ AB < 80 ≤ A
Bila belum lulus, ada:
• Ujian Reevaluasi (12 Mei 2017)
1/22/2014 6(c) Hendra Gunawan
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BAB 7. TEKNIK PENGINTEGRALANMA1201 MATEMATIKA 2A
1/22/2014 (c) Hendra Gunawan 7
dx
x
x24
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Sasaran Kuliah Hari Ini
7.1 Aturan Dasar Pengintegralan
Mengetahui bentuk integral baku dan dapatmengubah bentuk integral yang diberikan kebentuk integral dengan substitusi peubah
7.2 Pengintegralan Parsial
Menghitung integral dengan teknikpengintegralan parsial
1/22/2014 8(c) Hendra Gunawan
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Bagaimana menghitung…
1/22/2014 (c) Hendra Gunawan 9
dxx.ln
dx
x
x24
dxx.sec
dxxex .cos
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7.1 ATURAN DASARPENGINTEGRALAN
MA1201 MATEMATIKA 2A
1/22/2014 (c) Hendra Gunawan 10
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Beberapa Bentuk Integral Baku
1/22/2014 11(c) Hendra Gunawan
.1,1
1
rCr
xdxx
rr
.cossin Cxxdx
.sincos Cxxdx .tan1
1 1
2Cxdx
x
.||ln1 Cxdxx
.tansec2 Cxxdx
.cotcsc2 Cxxdx
.Cedxe xx
dst dst
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Mengubah Integral ke Bentuk Baku dengan Substitusi Peubah
Misal kita ingin menghitung integral
Dengan substitusi peubah: u = ex, du = ex dx, kita dapat mengubahnya ke bentuk integral baku, yang sudah kita kenal:
1/22/2014 12(c) Hendra Gunawan
2.
1
x
x
edx
e
.tantan1
1
1
11
22CeCudu
udx
e
e x
x
x
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Contoh/Latihan
1. Tentukan
Jawab: Misalkan u = x2. Maka …
1/22/2014 13(c) Hendra Gunawan
dxx
x
)(cos 22
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2. Tentukan
Jawab: Misalkan u = ...
1/22/2014 14(c) Hendra Gunawan
dx
x
x24
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3. Tentukan
Jawab:
1/22/2014 15(c) Hendra Gunawan
dx
x
xx4
3
1
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4. Tentukan
Jawab:
1/22/2014 16(c) Hendra Gunawan
dxee xx )tan(
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5. Buktikan kesamaan
lalu peroleh rumus berikut:
Jawab:
1/22/2014 17(c) Hendra Gunawan
.|tansec|lnsec Cxxxdx
,sin1
cos
cos
sinsec
x
x
x
xx
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7.2 PENGINTEGRALAN PARSIALMA1201 MATEMATIKA 2A
1/22/2014 (c) Hendra Gunawan 18
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Aturan Hasil Kali danPengintegralan Parsial
1/22/2014 19(c) Hendra Gunawan
b
a
b
a
b
a xduxvxvxuxdvxu
xduxvxvxuxdvxu
dxxuxvxvxudxxvxu
xvxuxvxudx
dxvxu
xvxuxvxuxvxudx
d
)()()]()([)()(
)()()()()()(
)(')()()()(')(
)()(')]()([)(')(
)()(')(')()]()([AturanHasil Kali:
Integralkankedua ruas:
Integral Tentu:
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Contoh/Latihan
1. Tentukan
Jawab:
1/22/2014 20(c) Hendra Gunawan
xdxx cos
)(sincos xxdxdxx
.cossin Cxxx
xdxxx sinsin
u = x, dv = cos x dxdu = dx, v = sin x
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2. Tentukan
Jawab:
1/22/2014 21(c) Hendra Gunawan
dxxex
)( xx exddxxe
...
u = x, dv = ex dxdu = dx, v = ex
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3. Tentukan
Jawab:
1/22/2014 22(c) Hendra Gunawan
dxex x2
)(22 xx edxdxex
...
u = x2, dv = ex dxdu = …, v = …
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4. Tentukan
Jawab:
1/22/2014 23(c) Hendra Gunawan
dxx.ln
u = ln x, dv = dxdu = … , v = …....ln dxx
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5. Tentukan
Jawab:
1/22/2014 24(c) Hendra Gunawan
dxxex .cos
u = … , dv = …du = … , v = …....cos dxxex
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6. Buktikan rumus berikut:
Lalu hitunglah
1/22/2014 25(c) Hendra Gunawan
dxxn
nxx
ndxx nnn .cos
1sin.cos
1.cos 21
..cos
2/
0
8
dxx