sifat khusus integral tentu
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SIFAT INTEGRAL TENTUMatematika Peminatan SMA
Sifat Khusus Integral Tentu
Aturan Kesimetrian
Untuk setiap π(π₯) genap maka
Untuk setiap π(π₯) ganjil maka
Aturan Keperiodikan
Jika π fungsi periodik dengan periode π maka
βπ
π
π π₯ ππ₯ = 2
0
π
π π₯ ππ₯
βπ
π
π π₯ ππ₯ = 0
π+π
π+π
π π₯ ππ₯ =
π
π
π π₯ ππ₯
Contoh
βπ
π
ππ β ππ π π
JawabMisal π π = ππ β ππ
π βπ = βπ π β π βπ = βππ + ππ = β ππ β ππ = βπ(π)
β΄ π π fungsi ganjil
βπ
π
ππ β ππ π π = 0
Contoh
βπ
π
ππππ
ππ π
Jawab
Misal π π = ππππ
π
π βπ = πππβπ
π= πππ
π
π= π(π)
β΄ π π fungsi genap
βπ
π
ππππ
ππ π = π
π
π
ππππ
ππ π = 2 4 sin
π₯
4 0
π
= 2 4 sinπ
4β 4 sin 0
= 2 4 β1
22 β 4 β 0 = 4 2
Contoh
π
ππ
πππ π π π
Jawab
Dari gambar diatas menunjukkan grafik π π = πππ π merupakan fungsi periodik dengan periode Ο, sehingga
π
ππ
πππ π π π =
π
π
πππ π π π +
π
ππ
πππ π π π
Y
X0 Ο 2Ο
π
ππ
πππ π π π =
π
π
πππ π π π +
π
ππ
πππ π π π
=
π
π
πππ π π π +
π+π
π +π
πππ π π π
=
π
π
πππ π π π +
π
π
πππ π π π
= π
π
π
πππ π π π
= 2 βcos π₯ 0π
= 2 βcos π + cos 0
= 2 β β1 + 1
= 4
Contoh
Jika βπππ π πππ π + π π π = π , dengan π(π) fungsi genap dan
βπππ π π π = π, maka βπ
ππ π π π adalah ...
Jawab
βπ
π
π π πππ π + π π π = π
βΊ
βπ
π
π π πππ π π π +
βπ
π
π π π π = π
π(π) fungsi genap dan sin π₯ fungsi ganjil maka π π πππ π fungsi ganjil
sehingga βπππ π πππ π π π = π dan βπ
ππ π π π = π π
ππ π π π.
Soal Matematika IPA SMBPTN 2017
Oleh karena itu
βΊ
βπ
π
π π πππ π π π +
βπ
π
π π π π = π
βΊ π+ π
π
π
π π π π = π
βΊ
π
π
π π π π = π
Sedangkan
βπ
π
π π π π = π βΊ
βπ
π
π π π π +
π
π
π π π π = π
βΊ
βπ
π
π π π π + π = π
βΊ
βπ
π
π π π π = π Soal Matematika IPA SMBPTN 2017
Contoh
Diketahui π(π) = π(π + π)untuk setiap π₯. Jika πππ π π π = π©, maka
πππ π + π π π = β―
Jawab
π
π
π π π π =
π
π
π π π π +
π
π
π π π π = π©
Misalkan πππ π π π = π¨maka π
ππ π π π = π© β π¨
π
π
π π + π π π =
π
π
π π π π
=
π
π
π π π π +
π
π
π π π π +
π
π
π π π π
=
π
π
π π + π π π +
π
π
π π + π π π +
π
π
π π + π π π
π
π
π π + π π π =
π
π
π π + π π π +
π
π
π π + π π π +
π
π
π π + π π π
=
π
π
π π π π +
π
π
π π π π +
π
π
π π π π
= π© β π¨ + π© + π¨
= ππ©
Soal Matematika IPA SMBPTN 2016
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