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hyperbolic functions

sarah12345

how do i intergrate cosh^2 x ?

thanks

Reply 1

steve2005

sarah12345

how do i intergrate cosh^2 x ?
thanks

Here is a solution....I have no idea how it is done.

Picture 286.png11.1KB

Reply 2

JaF

Use the double angle formula:

cosh(2x) = 2cosh^2 x -1

cosh^2 x = ½(1+cosh 2x)

intergrating gives:

½x + ¼sin2x

Reply 3

Aitch

sarah12345

how do i intergrate cosh^2 x ?
thanks

cosh 2x = 2cosh²x - 1
½cosh 2x + ½ = cosh²x
Integrate LHS
Get
¼sinh 2x + ½x

Aitch

[As produced earlier by JaF! :frown:

]

Reply 4

sarah12345

thanks i get it now!! also if tanh x = 1/2
how would you work out cosh x and sinh x ?

thanks alot

Reply 5

evariste_3086

sarah12345

how do i intergrate cosh^2 x ?

thanks

OR
cosh x=e^x+e^(-x)/2
cosh^2x= 1/4[e^(2x)+2+e^(-2x)]
int cosh^2(x)= 1/4[1/2e^(2x)+2x-1/2e(-2x)]+C
=1/4[e^(2x)-e^(-2x)/2]+x/2+C
=1/4sinh(2x)+x/2+C

Reply 6

Aitch

sarah12345

thanks i get it now!! also if tanh x = 1/2
how would you work out cosh x and sinh x ?

thanks alot

Start from 1 - tanh²x = sech²x...

Aitch

Reply 7

evariste_3086

sarah12345

thanks i get it now!! also if tanh x = 1/2
how would you work out cosh x and sinh x ?

thanks alot

tanhx=e^(x)-e^(-x)/e^(x)+e^(-x)=1/2
=> 2[e^(x)-e^(-x)]=e^(x)+e^(-x)
e^(x)-3e^(-x)=0
e^(2x)-3=0
e^(x)=rt(3)

Reply 8

Aitch

Start from 1 - tanh²x = sech²x...

Aitch

...then get cosh²x from sech²x, then sinh²x from cosh²x...

Aitch

Reply 9

Peevee

sarah12345

thanks i get it now!! also if tanh x = 1/2
how would you work out cosh x and sinh x ?

thanks alot

tanh x = (sinh x) / (cosh x) = 1/2

Hence, sinh[x] = 1 and cosh[x] = 2 by comparing, no?

Reply 10

Aitch

Peevee

tanh x = (sinh x) / (cosh x) = 1/2

Hence, sinh[x] = 1 and cosh[x] = 2 by comparing, no?

Um... How about sinhx=2 and coshx=4?

Aitch

[In]

Reply 11

Peevee

Aitch

Um... How about sinhx=2 and coshx=4?

Aitch

[In]

Ahh, u got me there! :biggrin:

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