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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

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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I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.**Education/Credentials**

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