This is an arc length integral. Apparently you wish to compute the length of that curve in the plane, with x going from 0 to pi/2. As a numerical integral, this is easy.
syms x
f=log(45.*sin(x.^2) + cos(x)); % initial function
fplot(f,[0,pi/2])
g=sqrt(1+diff(f).^2); % requirement step 1
gfun = matlabFunction(g);
First, it alway makes sense to plot everything, before you just throw it into any general tool.
fplot(g,[0,pi/2])
I don't epect that arc length integral has an exact solution. I might also be slightly worried about what seems to be a singularity at 0, though an integration tool should survive that. You can use vpaintegral, if you wish to stay in the symbolic domain.
vpaintegral(g,[0,pi/2])
Or you can use integral.
integral(gfun,0,pi/2)
Which seems to agree.
And, since I have a tool on the file exchange that can compute the arclength of a curve, I might try this:
ffun = matlabFunction(f);
xint = linspace(0,pi/2);
arclength(xint,ffun(xint),'spline')
ans =
4.7383
which also seems to agree pretty well.
