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# Calculus/Hyperbolic functions

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Question
I am a scientist and have a problem I'm working by hand and messed up somewhere, but not sure where.

The problem is the integral of x^2/Sqrt(x^2+2)dx from 0 to 5.

I began by letting x=sqrt(2)sinh(t) so dx=sqrt(2)cosh(t)dt. So the integrand is 2*Integral[sinh(t)^2dt]. Doing the integral, I get arcsinh(x/sqrt[2]]-(x/2)(sqrt[x^2+2)). Evaluating the definite integral, I get -11.015, but my calculator gives me 11.015. I don't see where I picked up the negative. Any ideas?

Thanks

Answer
Questioner:David
Country:Oklahoma, United States
Category:Calculus
Private:Yes <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< changed.

I am a scientist and have a problem I'm working by hand and messed up somewhere, but not sure where.

The problem is the integral of x^2/Sqrt(x^2+2)dx from 0 to 5.

I began by letting x=sqrt(2)sinh(t) so dx=sqrt(2)cosh(t)dt.
So the integrand is 2*Integral[sinh(t)^2dt].
Doing the integral, I get arcsinh(x/sqrt[2]]-(x/2)(sqrt[x^2+2)).
Evaluating the definite integral, I get -11.015, but my calculator gives me 11.015.
I don't see where I picked up the negative. Any ideas?

Thanks
----------------------------------------------
Running your integral through THE INTEGRATOR (not a former governor of california, a

computer program) gives:

(x*Sqrt[2 + x^2])/2 - ArcSinh[x/Sqrt[2]]

which is the opposite of your integral.  So you got something backwards in the integrating process:

Note:  I write 's2'  to mean  sqrt(2); saves typing.

I think you want to handle  x^2 + 2  by substituting :
x = s2 sinh t
sqrt(2 + x^2) = s2 cosh t
dx = s2 cosh t

x^2
------------- dx =
sqrt(x^2 + 2)

2 sinh^2 t s2 cosh t dt
----------------------------- =
s2 cosh t

2 sinh^2 t dt  =

Use a 'double-angle' identity:
cosh 2x = sinh^2 x + cosh^2 x = 2 sinh^2 x + 1

2 sinh^2 t dt  =  cosh 2t - 1   <<<<< is this where you blew the sign?

Integrate:

1/2 sinh 2t - t

Another identity: sinh 2t = 2 sinh t cosh t   gives:

sinh t cosh t - t

Back substiute for t:

(x / s2)(sqrt(x^2 + 2)/s2) -  arcsinh(x/s2)

x sqrt(x^2 + 2)/2 - arcsinh(x/s2)

You can look through and see if you can find the error now.

Calculus

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