numerical integration and solving for limit

Question

PTK 2023-1-19

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评论： Walter Roberson 2023-1-20

hello everyone, this is my first question here.

I'm trying to find the value of x when t equals 1:0.1:10

So I don't know how to solve this problem, for example if the intervals were fixed and I wanted to find t, I would just use the commands

f(x)= (-1./(sqrt((1-x.^2)+(0.01/2)*(1-x.^4))))

a=1;

b=5;

I=int(f(x),a,b);

Can anybody help me?

how do i find the value of b when t is equal to 1:0.1:10.

Thanks for your help.

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Answer 1

Torsten 2023-1-19

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编辑：Torsten 2023-1-19

在 MATLAB Online 中打开

Here is the maximum value you can insert for t:

syms x
f = 1/sqrt((1-x^2)+0.01/2*(1-x^4));
vpaintegral(f,x,-1,1)
ans = 3.12988

And here is the curve of t against x.

You can cheat here: plot x against t and say you solved t = -integral ... for x. :-)

xstart = -1;

xend = 1;

xnum = linspace(xstart,xend,100);

for i=1:numel(xnum)

tnum(i) = double(vpaintegral(f,x,xnum(i),1));

end

plot(xnum,tnum)

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Torsten 2023-1-19

Yes, I plotted x against the value of the integral.

PTK 2023-1-19

Thank you so much!

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Answer 2

Walter Roberson 2023-1-19

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your f has x^4 and x^2 but no other powers of x.

Do a change of variables X2=x^2 and integrate with respect to X2. You will get a closed form integral involving arcsin. Transform back from X2 to x. Now you can solve the equation. Just make sure you get the right limits of integration

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PTK 2023-1-19

Thank you.

Walter Roberson 2023-1-20

The problem with this approach turns out to be that you would need x^2 to be negative to get at some of the values, which is a problem because that gets you into complex-valued x.

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