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definition of even and odd functions from 1.2
An even function f satisfies f(x)=f(x), while an odd function satisfies f(x)=f(x)

Every functin f that is defined on an interval centered at the origin can be written in a unique way as the sum of one even functin and one odd functin.. WHAT is the composition?? What is the composition written e^x?
f(x)=f(x)+f(x)/2 + f(x)  f(x)/2 e^x = e^x + e^x/2 + e^x  e^x/2

what is the definition of hyperbolic funtions?
The even and odd parts of e^x, and are called hyerbolic cosine and hyperbolic sine of x

what is hyperbolic sine of x??
sinhx = e^x  e^x/2

what is the hyperbolic cosine of x?
coshx = (e^x + e^x)/2

tanh x?
Sinhx/coshx = (e^x  e^x)/e^x +e^x

coth x?
= cosh x/sinh x = (e^x + e^x)/e^x  e^x

Hyperbolic secant?
sech x = 1/coshx = 2/(e^x + e^x)

hyperbolic csc?
csch x + 1/sinhx = 2/(e^x  e^x)

know the identities for these 7 hyperbolic functions 1. cosh^2 x  sinh^2 x
2. sinh 2x
3. cosh 2x
4. cosh^2 x
5. sinh^2 x
6.tanh^2 x
7coth^2 x
 1=1
 2=2sinh x cosh x
 3= cosh^2 x + sinh^2 x
 4=(cosh 2x + 1)/2
 5=(cosh 2x 1)/2
 6 = 1  sech^2 x
 7 = 1 + csch^2 x

what are the derivatives of the six hyperbolic functions? d/dx of?
1 sinh u?
2 cosh u?
3 tanh u?
4 coth u?
5 sech u?
6 csch u?
 1 cosh u du/dx
 2 sinh u du/dx
 3 sech ^2 u du/dx
 4 csch^2 u du/dx
 5 sech u tanh u du/dx
 6 csc u coth u du/dx

the derivatives of the hyperbolic functions lead to the formulas for the integrals of hyperbolic fnctions. what are the integrals of these six hyperbolic functions??
1 sinh u du?
2 cosh u du?
3 sech^2 u du?
4 csch^2 u du?
5sech u tan u du?
6 csch u coth u du?
 1 cosh u + C
 2 sinh u + C
 3 tanh u + C
 4 coth u + C
 5 sech u + C
 6 csch u + C

There are also six derivatives of inverse hyperbolic functins d (____^1 u)/dx. What are they?
1 sinh?
2 cosh ?
3 tanh ?
4 coth ?
5sech?
6 csch?
 1= 1/ sqrt(1 + u^2) du/dx
 2 = 1/sqrt(u^2 1) du/dx, u>1
 3 = 1/sqrt(1  u^2) du/dx, u < 1
 4 = 1/sqrt(1 u^2) du/dx, u > 1
 5 = du/dx/ u sqrt(1u^2), 0< u< 1
 6 = du/dx/ u sqrt (1 + u^2), u cant = 0

now shit gets a little crazy. there are 5 integrals that lead to inverse hyperbolic functions.
1 fdu/ sqrt(a^2 + u^2)
2 fdu/ sqrt(u^2  a^2)
3 fdu/ sqrt(a^2  u^2)
4 fdu/ usqrt(a^2  u^2)
5 fdu/ usqrt(a^2 + u^2) what are the inverse hyperbolic functions?
 1 = sinh^ 1 (u/a) + C a > 0
 2= cos^1 (u/a) + C u > a > 0
 3= 1/a tanh^1 (u/a) + C if u^2 < a^2 or 1/a coth^1 (u/a) + C if u^2 > a^2
 4= 1/a sech^1 (u/a) + C, 0 < u < a
 5=1/a csch^1  u/a  + C, u cant = 0 and a>0

three identities for inverse hyperbolic functions
1 sech^1 x
2 csch^1x
3 coth^1 x
 1= cosh^1 1/x
 2= sinh^1 1/x
 3=tanh^1 1/x

