integration

please can someone help me to solve this problem: I want to integrate this function:

exp(0.5*(1-((l-hm)/(2*H))-exp(-((l-hm)/(2*H)))))

I make this program:

hm=400;
H=50;
syms l;
H0=int(exp(0.5*(1-((l-hm)/(2*H))-exp(-((l-hm)/(2*H))))),l,hm,700)

but the answer is:

100*2^(1/2)*pi^(1/2)*exp(1)^(1/2)*(erf(2^(1/2)/2) - erf(2^(1/2)/(2*exp(3/2))))

how to get the right answer???

Answer 1

[Alexander]

Why do you think that the answer is incorrect? If you just want a number, use double on the result:

>> double(H0)

ans =

    209.1673

Or use INTEGRAL as Mike Hosea suggests.

[Mike Hosea]

:lol: I didn't even bother with the symbolic attempt and completely misinterpreted the question. I'm blaming this on Walter. :)

[Walter Roberson]

It's a fair cop, gov!

Answer 2

[Mike Hosea]

I never quite know what to say when questions like this come up. There's nothing wrong with wanting to do integrals analytically, to be sure, and some times there are legitimate reasons, I guess, but I wonder what people need the analytical answer for. This problem is really no problem for INTEGRAL (or QUADGK)

>> integral(@(l)exp(0.5*(1-((l-hm)/(2*H))-exp(-((l-hm)/(2*H))))),hm,700)
ans =
     2.091673274741571e+02
>> 100*2^(1/2)*pi^(1/2)*exp(1)^(1/2)*(erf(2^(1/2)/2) - erf(2^(1/2)/(2*exp(3/2))))
ans =
     2.091673274741571e+02

Answer 3

[Walter Roberson]

MuPAD is not able to complete that integral. (Maple is not able to do it either.)

Perhaps there is an appropriate change of variables, but I have not been able to locate one.

[Alexander]

I believe the result is correct. Converting it to a double gives the same result as the numerical integration.

Why do you think that MuPAD cannot compute it?

[Walter Roberson]

MuPAD's integration is much weaker than Maple's, so if Maple cannot integrate it then MuPAD almost certainly cannot (at least not without a lot of step by step work.)

The result you list might be correct, but I have not found a way to obtain it.

Mathematica is also not able to integrate it, at least within the default timeout of their free service,

http://www.wolframalpha.com/input/?i=integrate+exp%280.5*%281-%28%28l-400%29%2F%282*50%29%29-exp%28-%28%28l-400%29%2F%282*50%29%29%29%29%29+dl+l%3D400..700

[Christopher Creutzig]

Walter, what do you mean, “MuPAD is not able to complete that integral”? It obviously is, Edward even showed the answer MuPAD returns. I have to agree with Alexander, the result seems to be completely correct.

[Alexander]

Just to be sure, we are talking about the same thing: MuPAD does in fact give the above answer. It's also able to calculate the indefinite integral:

int(exp((1 - (l - 400)/(2*50) - exp(-(l - 400)/(2*50)))/2), l)

-100*2^(1/2)*PI^(1/2)*exp(1)^(1/2)*erf((2^(1/2)*exp(2)*exp(-l/200))/2)

This can be verified by differentiation.

[Walter Roberson]

I think I misunderstood the original posting.

[Christopher Creutzig]

If it helps to get some confidence in the result: To get integrals.com to integrate the expression (indefinitely), substitute x=exp(-l/200). http://integrals.wolfram.com/index.jsp?expr=-200*exp%285%2F2+-+%281%2F2%29*x^2*exp%284%29%29&random=false

[Alexander]

Okay, that explains my confusion.