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Integrating inverse hyperbolic functions

generalebriety

dy/dx = arsinh x

I'm completely stuck. :confused:

Any ideas?

Reply 1

Speleo

Hint: how would you integrate lnx?

Reply 2

generalebriety

Ooh, good thinking. :smile:

Thanks.

Edit: so just to check (someone confirm please?), y = x arsinh x - ln sqrt(x^2+1) + c?

Reply 3

Speleo

That is correct, darth.

generalebriety: x is not the derivative of sqrt(1 + x^2), you didn't apply the chain rule properly :tongue:

Reply 4

generalebriety

Speleo

That is correct, darth.

generalebriety: x is not the derivative of sqrt(1 + x^2), you didn't apply the chain rule properly :tongue:

I didn't apply the chain rule at all. :confused:

dy/dx = arsinh x
Integrating by parts,
u = arsinh x, u' = 1/sqrt(x^2 + 1)
v' = 1, v = x

y = x arsinh x - INT x/sqrt(x^2 + 1) dx
Let w^2 = x^2 + 1; then w dw = x dx.
y = x arsinh x - INT w/w^2 dw
= x arsinh x - INT 1/w dw
= x arsinh x - ln w + c
= x arsinh x - ln sqrt(x^2 + 1) + c.

What have I done wrong?

Edit: ignore me, found it. Thanks for the help. :smile:

Reply 5

Speleo

I assumed that you'd ended up with INT x/sqrt(1+x^2) dx and said that d/dx sqrt(1+x^2) = 1/2 * 2x and ignored the sqrt(1+x^2)^(-1/2).

Reply 6

insparato

I think the most likely error was that, put w^2 on the bottom when in actual fact its w.

Reply 7

generalebriety

insparato

I think the most likely error was that, put w^2 on the bottom when in actual fact its w.

Well, thank you for your input into the situation, insparato. :p:

Reply 8

insparato

I made that error myself hey im not trying to sound superior here believe me im a far worst mathematician.

Reply 9

Speleo

I think the most likely error was that, put w^2 on the bottom when in actual fact its w.

Well that is obvious now he has posted his method :tongue:

I was explaining my post after the incorrect answer and before the working.

Reply 10

generalebriety

insparato

I made that error myself hey im not trying to sound superior here believe me im a far worst mathematician.

Whether that's true or not, it doesn't stop you taking the piss from time to time. :p:

Keeps me on my toes, though. :biggrin:

If anyone's interested, that question was taken from STEP I 1992 Q2.

Reply 11

insparato

Oh right that can happen.

Reply 12

insparato

generalebriety

Whether that's true or not, it doesn't stop you taking the piss from time to time. :p:

Keeps me on my toes, though. :biggrin:

If anyone's interested, that question was taken from STEP I 1992 Q2.

wasnt trying to take the piss this time honest. My post was full of sincerity and no piss taking intentions.

Reply 13

generalebriety

insparato

wasnt trying to take the piss this time honest. My post was full of sincerity and no piss taking intentions.

Fair enough, sorry. :biggrin:

Reply 14

insparato

Not a problem :p:

. (Interesting STEP question though).

Reply 15

generalebriety

Another integration question I can't do (not hyperbolic, but can't be bothered to make another thread): INT 1/(3cosx + 4sinx) dx. Any ideas? I've tried dividing top and bottom by cosx but that doesn't help, I've tried every single substitution I know... bah. :p: