Hyperbolic Functions

Question 1

sinh(x)

Answer 1

“shine(x)”
(e^x - e^-x) /2

Question 2

cosh(x)

Answer 2

“cosh(x)”
(e^x + e^-x) /2

Question 3

tanh(x)

Answer 3

(e^2x - 1)/(e^2x + 1)

Question 4

What are the reciprocal hyperbolics and draw the graphs.

Answer 4

cosech(x)
sech(x)
coth(x)
1/the original like in trig

Question 5

Proving inverse hyperbolic formulae

Answer 5

1. Use the hyperbolic on both sides to write in terms of x
2. Replace with the hyperbolic formula
3. Form a hidden quadratic by multiplying by e^y
4. ln both sides, only use the positive case
5. Replace y with the inverse hyperbolic

Question 6

What are the graphs of all the inverse hyperbolic functions?

Answer 6

Question 7

Hyperbolic pythagorean identities

Answer 7

cosh^2x - sinh^2x = 1
sech^2x = 1 - tanh^2x
cosech^2x = coth^2x - 1

Question 8

sinh(A+/-B)

Answer 8

sinhAcoshB +/- coshA/sinhB

Question 9

cosh(A+/-B)

Answer 9

coshAcoshB +/- sinhAsinhB

Question 10

tanh(A+/-B)

Answer 10

tanhA -/+ tanhB / 1 +/- tanhAtanhB

Question 11

Osborn’s rule

Answer 11

Replace sin with sinh and cos with cosh
Put a - in front of any multiplication of sinhx (including tanh^2) (sinh^4 cancels out)

Question 12

d/dx(coth(x))

Answer 12

-cosech^2(x)

Question 13

d/dx(sinh(x))

Answer 13

cosh(x)

Question 14

d/dx(cosh(x))

Answer 14

sinh(x)

Question 15

d/dx(tanh(x))

Answer 15

sech^2(x)

Question 16

Proving regular differentials

Answer 16

Put it into exponential form and differentiate

Question 17

Proving inverse derivatives

Answer 17

1. x = sinh/cosh/tanh y
2. Do dx/dy
3. Use identities to substitute in terms of x
4. Find the reciprocal for dy/dx

Question 18

∫sinh(x) dx

Answer 18

cosh(x) + c

Question 19

∫cosh(x) dx

Answer 19

sinh(x) + c

Question 20

∫tanh(x) dx

Answer 20

sech^2(x) + c

Question 21

∫cosech^2(x) dx

Answer 21

-coth(x) + c

Question 22

∫sech^2(x) dx

Answer 22

tanh(x) + c

Question 23

∫sech(x)tanh(x) dx

Answer 23

-sech(x) + c

Question 24

∫cosech(x)coth(x) dx

Answer 24

-cosech(x) + c

Question 25

Multiple terms in the numerator

Answer 25

Split and use standard integrals/reverse chain

Question 26

Proving ∫tanh(x)

Answer 26

use sinh(x)/cosh(x) and use reverse chain rule to ln|cosh(x)|

Question 27

Small odd powers of cosh/sinh

Answer 27

Factor out (cosh/sinh)^power-1 and use identity

Question 28

When to use the exponential definition

Answer 28

When there is an exponential term or no simpler way to integrate

Question 29

∫sech(x) or ∫cosech(x) method

Answer 29

Use exponential form, multiply by e^x and use substitution with e^x

Question 30

∫1/sqrt(a^2 + x^2) substitution

Answer 30

x = asinh(u)

Question 31

∫1/sqrt(x^2 - a^2) substitution

Answer 31

x = acosh(u)

Question 32

Completing the square

Answer 32

With a/quadratic or a/sqrt(quadratic), complete the square and use substitution with u as the thing that is squared

Question 33

cosh^2(x) substitution

Answer 33

1/2 + 1/2 cosh(2x)

Question 34

sinh^2(x) substitution

Answer 34

1/2 cosh(2x) - 1/2

Question 35

Hyperbolics to R formula

Answer 35

Use cosh^2 - sinh^2 = 1 for R rather than +