Symbolic Integration Help
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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
采纳的回答
Walter Roberson
2011-1-23
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.
5 个评论
Walter Roberson
2011-1-23
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
2011-1-24
Walter Roberson
2011-1-25
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
2011-1-25
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
2011-1-26
>> 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)
更多回答（1 个）
Paulo Silva
2011-1-23
0 个投票
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).
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2011-1-23
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