Symbolic Integration Help

I want to integrate this function, with respect to x...

y=1/((1+x^2)*(1+x^2+B1^2)^.5)

and B1 is a changing variable. If B1 is a constant, I carried out the symbolic integration like this...

y=sprintf('1/((1+x^2)*(1+x^2+(%1.3f)^2)^.5)',B1); s=int(y,z1,z2);

where z1 and z2 are my limits of integration and it works. How can I perform this integration when B1, z1, and z2 are changing? I tried doing a for loop, but it didn't work.

for x=1:100 y(x)=sprintf('1/((1+x^2)*(1+x^2+(%1.3f)^2)^.5)',B1(x)); s(x)=int(y(x),z1(x),z2(x)); end

Answer 1

[Walter Roberson]

The complete answer is messy because you have not specified that B1, z1,or z2 are real, or that z1 <= z2. If you make those assumptions, you get

|

                  /      /              2          \
              1   |      |         z1 B1           |
           - ---- |arctan|-------------------------|
             |B1| |      |               (1/2)     |
                  |      |/      2     2\          |
                  \      \\1 + z1  + B1 /      |B1|/
                      /              2          \\
                      |         z2 B1           ||
              - arctan|-------------------------||
                      |               (1/2)     ||
                      |/      2     2\          ||
                      \\1 + z2  + B1 /      |B1|//

|

I will post later if I find a more compact solution.

[Walter Roberson]

It simplifies a bit to

-(-arctan(z1*B1/sqrt(1+z1^2+B1^2))+arctan(z2*B1/sqrt(1+z2^2+B1^2)))/B1

under the assumption that the values involved are real.

[Devin Rohan]

Thanks, I figured out how to do integrate it symbolically, but it is really slow. I think I'll have to use that solution you came up with. How did you find that solution? When I typed in syms x B1 z1 z2 s=int(1/((1+x^2)*sqrt(1+x^2+B1^2)),z1,z2), MATLAB said it couldn't find a solution.

[Walter Roberson]

I used Maple, and I told Maple to assume real values and to assume z1<=z2 . That got me the less compact solution in a small number of seconds. To make it more compact, I ran it through a number of potential transformations and picked the most compact answer out of them.

In this particular case, when I told Maple to rewrite in terms of Heaviside functions and then to simplify the answer, it was able to simplify the B1^2 and abs(B1) components. I do not entirely trust Maple's Heaviside conversions, but Maple's conversion to signum provided an answer that could obviously be simplified to the one Heaviside came up with.

[Walter Roberson]

By the way, see the "real" modifier of "syms" to allow you to add the assumption that a particular symbolic variable is real.

[Christopher Creutzig]

>> syms x B1 z1 z2 real

>> s=int(1/((1+x^2)*sqrt(1+x^2+B1^2)),z1,z2)

s =

-(atan((z1*(B1^2)^(1/2))/(B1^2 + z1^2 + 1)^(1/2)) - atan((z2*(B1^2)^(1/2))/(B1^2 + z2^2 + 1)^(1/2)))/(B1^2)^(1/2)

Answer 2

[Paulo Silva]

Use the function syms to declare symbolic variables and subs to replace a variable with a specific value, you might also need to use the function vectorize to convert symbolic expressions to char (just in case you want to plot something).