1.5 FUNGSI HIPERBOLIK
1.5.1 Definisi fungsi hiperbolik
1. Sinus hiperbolik : sinh x =
2 e
e x
x −
− 2. Cosinus hiperbolik :
cosh x = 2
e e
x x
− +
3. Tangent hiperbolik : tanh x =
x cosh
x sinh
= x
x x
x e
e e
e +
− −
4. Cotangent hiperbolik : coth x =
x sinh
x cosh
= x
x x
x e
e e
e −
+ − 5. Secant hiperbolik :
sech x = x
cosh 1
= x
x e
e 2
− +
6. Cosecant hiperbolik : csch x =
x sinh
1 =
x x
e e
2 −
−
Persamaan dasar mirip dengan fungsi trigonometri biasa: Fungsi Hiperbolik
Fungsi Trigonometri a. tanh x =
x coth
1 tan x =
x cot
1 b. cosh
2
x – sinh
2
x = 1 cos
2
x + sin
2
x = 1 c. 1 – tanh
2
x = sech
2
x 1 + tan
2
x = sec
2
x d. 1 – coth
2
x = – csch
2
x 1 + cot
2
x = csc
2
x Tugas : Buktikan
1. cosh x + sinh x = e
x
6. cosh 2x = cosh
2
x + sinh
2
x 2. cosh x – sinh x = e
-x
7. sinh 2x = 2 sinh x cosh x
3. 2
1 x
cosh x
2 1
sinh2 −
= 8. sinh x + y = sinh x cosh y + cosh x sinh y
4. tanh 2x =
x tanh
1 x
tanh 2
2
+
9. cosh x + y = cosh x cosh y + sinh x sinh y 5.
2 1
x cosh
x 2
1 cosh2
+ =
1.5.2 Turunan Fungsi Hiperbolik
a. Fungsi y = sinh x = 2
e e
x x
− −
, turunannya dx
dy =
2 e
e x
x −
+ = cosh x
b. Fungsi y = cosh x = 2
e e
x x
− +
, turunannya dx
dy =
2 e
e x
x −
− = sinh x
c. Fungsi y = tanh x = x
x x
x e
e e
e +
− − , turunannya
dx dy
=
2 x
x
e e
2
+
−
= sech
2
x d. Fungsi y = coth x =
x x
x x
e e
e e
− +
−
, turunannya dx
dy =
2 x
x
e e
2
−
−
− = – csch
2
x e. Fungsi y = sech x =
x x
e e
2
−
+
, turunannya dx
dy = –
x x
e e
2
−
+
x x
x x
e e
e e
− −
+ −
= – sech x tanh x f.
Fungsi y = csch x =
x x
e e
2
−
−
, turunannya dx
dy =
2 x
x x
x
e e
e e
2
− −
− +
−
= – csch x coth x – 2
2 X
Y
y = sinh x X
Y y = cosh
x
– 1 1
X Y
y = tanh x
Grafik fungsi y = sinh x, y = cosh x, dan y = tanh x
Secara umum: a. y = sinh u , turunannya
dx dy
= cosh u dx
du b. y = cosh u, turunannya
dx dy
= sinh u dx
du c. y = tanh u, turunannya
dx dy
= sech
2
u dx
du d. y = coth u, turunannya
dx dy
= – csch
2
u dx
du e. y = sech u, turunannya
dx dy
= – sech u tanh u dx
du f.
y = csch u, turunannya dx
dy = – csch u coth u
dx du
Contoh soal: Tentukan turunan dari 1. y = tanh 1 – x
2
Jawab : dx
dy = – 2x sech
2
1 – x
2
2. y = ln sinh x Jawab :
dx dy
= x
sinh x
cosh = coth x
3. y = tanh 5
1 x
4 +
Jawab : dx
dy =
5 1
x 4
h sec
5 4
2
+ Tugas : Tentukan turunan dari
1. y = x sech x
2
4. y = csch
2
x
2
+ 1 2. y = ln cosh x
5. y = a cosh a
x 3. y =
1 x
tanh 1
+ 1.5.3
Integrasi Fungsi Hiperbolik
Rumus-rumus pokok integrasi fungsi hiperbolik a. sinh u du = cosh u + C
b. cosh u du = sinh u + C c. tanh u du = ln | cosh u | + C
d. coth u du = ln | sinh u | + C e. sech
2
u du = tanh u + C f.
csch
2
u du = – coth u + C g. sech u tanh u du = – sech u + C
h. csch u coth u du = – csch u + C Contoh soal : Hitung integral berikut
1. ∫
dx x
h sec
= ∫
x cosh
dx =
∫ x
cosh dx
x cosh
2 =
∫ +
x sinh
1 dx
x cosh
2 misal u = sinh x maka du = cosh x dx, sehingga
= ∫
+ 2
u 1
du = arc tan u + C = arc tan sinh x + C
2. ∫
dx x
cosh e
x
= dx
2 e
e e
x x
x −
+ ∫
= ∫
+ dx
1 e
2 1
x 2
= 2
1 e
4 1
x 2
+ + C
Tugas : Hitung integral berikut 1.
∫ dx
x 2
1 cosh3
4. ∫
dx x
sinh x
2
2. ∫
dx x
h sec
4
5. ∫
dx x
sinh e
x
3. ∫
dx x
sinh x
6. ∫
dx x
cosh x
sinh
2 3
1.6 FUNGSI INVERSI HIPERBOLIK
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