Solving an equation with integral with one variable

Hi, how can I calculate the following equation involving an integral in matlab?

C + Integral_{-4}_{4} e^(x^2)*x^2 dx = 1

where -4 and 4 are the lower and upper limit, and C is the unknown constant.

Thanks!

Answer 1

[John BG]

Hi Sergio

this is John BG <mailto:[e-mail] [e-mail]>

Since the primitive of

    exp(x^2)*x^2

is

Ly=x*exp(x^2)/2-.5*1j*(pi)^.4/2*erf(1j*x)

then the integral in the interval [y1 y2] is

L=Ly2-Ly1

this is

y2=4;
Ly2=y2*exp(y2^2)/2-.5*1j*(pi)^.5*erf(sym(1j*y2))

Ly2 =
149084195602433/8388608 - (erf(4i)*3991211251234741i)/4503599627370496

   y1=-4;
  Ly1=y1*exp(y1^2)/2-.5*1j*(pi)^.5*erf(sym(1j*y1))

Ly1 =
(erf(4i)*3991211251234741i)/4503599627370496 - 149084195602433/8388608

the value of the integral is

   L= Ly2-Ly1
  L =
  149084195602433/4194304 - (erf(4i)*3991211251234741i)/2251799813685248

way around erf() only working for real inputs

L= double(Ly2-Ly1)
L =
   3.7843e+07

therefore

    C=1-L
    C =
      -3.7843e+07

Repeating with

format double

the result is

L =
     3.784324335121135e+07

Comment:

When attempting direct integration with command sum

    x=[-4:.001:4];L=sum(exp(x.^2) .* x.^2)
    L =
       3.4537e+10
    x=[-4:.00001:4];L=sum(exp(x.^2) .* x.^2)
    L =
       3.4396e+12
    x=[-4:.0000001:4];L=sum(exp(x.^2) .* x.^2)
    L =
       3.4395e+14

The slopes when approaching -4 and 4 are too sharp for command sum.

Grazie tante per la sua attenzione.

if you find this answer useful would you please be so kind to consider marking my answer as Accepted Answer?

To any other reader, if you find this answer useful please consider clicking on the thumbs-up vote link

thanks in advance

John BG

<mailto:[e-mail] [e-mail]>

Answer 2

[Star Strider]

If I understand correctly what you want to do, this works:

f = @(x) exp(x.^2) .* x.^2;
Cs = fzero(@(C) C + integral(f, -4, 4) - 1, 1)

Cs =
  -3.4395e+07

Or more simply, since ‘C’ is a constant:

C = 1 - integral(f, -4, 4)

C =
  -3.4395e+07

If you are using R2011b or earlier, use quad instead of integral:

C = 1 - quad(f, -4, 4)

The result is the same:

C =
         -34395040.4474417

here using format long g.

[Sergio Manzetti]

Hi Star Strider, I was thinking of something real simple, like on Wolfram Alpha, where you type in "C*integral of f(x) + integral g(x) = 1" . Then you get the output, for the numerical integration of the two integrands, and you get the value of C automatically. Is this feasible in Matlab?

[Star Strider]

Not to my knowledge.