Integrate x^2 /square root (x^2 +1)

[yanjt]

Hi,
the question is integrate x^2 /square root (x^2 +1).

I tried to do it but i was stucked in the very last step.at the end,I got x/2 * cosh(arcsinh x) + 1/2 * arcsinh x - arcsin (square root (x^2 + 1)) +C. I dunnoe how can I simplify cosh(arcsinh x) into simplier form in term of x. I hope what I wrote is not too confusing.Thanks!

 

[Dick]

I haven't checked your integration but arcsinh(z)=ln(z+sqrt(1+z^2)). You can put that into cosh(y)=(e^y+e^(-y))/2 and get an algebraic form for cosh(arcsinh(z)).

 

[tiny-tim]

Hi yanjt!

(have a square-root: √ and try using the X² tag just above the Reply box )

i] cosh(arcsinh x) is cosh of the "angle" whose sinh is x … so just use cosh² - sinh² = 1.

ii] why is your result so complicated? just use a substitution.

 

[yanjt]

Thanks, dick n tiny-tim!

I did it in this way:

x²/√(x²+1) = √(x²+1) - 1/√(x²+1)

Integrate 1/√(x²+1) = arcsinh x
For √(x²+1),i subsitute x=sinh t.
So, integrate √(x²+1 = cosh²t dt
= 1/4 sinh 2t + t/2
=1/4 (2 sinh t cosh t) + arcsinh x/2
=x/2 (cosh(arcsinh x)) + arcsinh x/2

 

[tiny-tim]

Hi yanjt!

Wouldn't it have been easier to substitute x = sinht at the very start?

(and you can't leave cosh(arcsinh x) as it is … you must convert that into something more "algebraic" )

 

[yanjt]

ooohhhh yeah!y i didnt think of tat? tat's a faster n easier way to do it!thanks! but is integrate 1/cosh x = 1/sinh x?can i jus integrate like tat?

i know tat i can't jus leave cosh(arcsinh x) as my final answer.tat's y i m stucked in this step. but i will try to change it into somethg more "algebraic" using the methods tat dick n u mentioned in the first 2 posts.