MATLAB: Solving the following integral in MatLab

numerical integration

Can anyone please help me to solve the following integration in MatLab.

int(log2(1+x)*exp(-c*x/(a-b*x)),x,0,inf)

I tried in Matlab and Mathematica. But the following results is found int((log(x + 1)*exp(-(c*x)/(a – b*x)))/log(2), x, 0, Inf)

and Mathematica shows the following message, Integrate::idiv: Integral of E^((c x)/(-a+b x)) Log[1+x] does not converge on {0,\[Infinity]}. >>

Is there anyway to solve this?

Thanks in advance.

Best Answer

If b and c are both nonzero, then as x approaches infinity the factor exp(-c*x/(a-b*x)) approaches exp(c/b) which would be nonzero and log2(1+x) approaches infinity. It is therefore obvious that the integral does not converge at the upper limit as x approaches infinity, since the integrand approaches infinity. You should not be asking how to solve this integral. You should be looking into whatever reasoning led to the integral and reconsider that reasoning if you expected a finite answer.

Related Solutions

MATLAB: Solution for integration of following expression.

A general form for the indefinite integral of your problem does not exist.

Take y = exp(a*x) and transform the integral over x to an integral over y. It will be the integral of

 1/a* 1/(1+b*y^(c/a)) *dy

depending on what the value of c/a is, a general form for the integral may/may not exist. For example, for c/a=1, the result is

 1/(a*b)* log(1+b*y)

but for c/a=2, b>0, the integral will be

 sqrt(b)/a*Arctan(sqrt(b)*y)

So I don't think you can get a general form of the integral from the Symbolic Math Toolbox or any other Symbolic Math Software. You can use the numerical integrations methods and integrate it over a definite domain.

MATLAB: Improper integral

I don't think quad ever allowed infinite limits. quadgk does. The answer you need depends on how old your MATLAB version is. I am going to express an answer in more recent syntax. If you try

a=0; b=1; 
fun = @(u) exp(-a.* exp(2.*u)).*exp(-(u + b.^2).^2./2.*b.^2)/...
  (sqrt(2.*pi.*b.^2));
quad(fun,-Inf,Inf)

then you get this message:

Warning: Infinite or Not-a-Number function value encountered.

and the answer is NaN. The likely reason is that there is underflow in these exponential expressions:

>> fun(1000)
ans =
   NaN

However, if you plot this function, you can see that this function approaches zero very rapidly. For example,

>> fun(0)
ans = 
    0

Therefore, you can do this calculation:

quadl(fun,-100,100)
ans = 
    1.0000

This is 1 to an accuracy of about 10^-8.

Note: the fun = ... line is an anonymous function.

Best Answer

Related Solutions

MATLAB: Solution for integration of following expression.

MATLAB: Improper integral

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