How to find the points of discontinuity af a function on interval?
Application
Theorem(Generalized Newton-Leibnitz formula).
Let F = integrate(f,x) and
x[1], x[2], ..., x[n] discontinuity points of F from interval [a, b],
p is list: p=[a, x[1], x[2], ..., x[n], b]. Then
integrate(f,x,a,b)=sum(F(p[k+1]-0)-F(p[k]+0),k,1,n+1)
Example. integate(cos(3*x)/(5-4*cos(x)),x, 0, 10*%pi)
(%i1) f:cos(3*x)/(5-4*cos(x))$
(%i2) F:integrate(f,x)$
discontinuity points of F is %pi, 3*%pi, 5*%pi, 7*%pi, 9*%pi
(%i3) p:[0,%pi,3*%pi,5*%pi,7*%pi,9*%pi,10*%pi];
(%o3) [0,%pi,3*%pi,5*%pi,7*%pi,9*%pi,10*%pi]
(%i4)
'integrate(f,x,0,10*%pi)=sum(limit(F,x,p[k+1],minus)-limit(F,x,p[k],plus),k,1,6);
(%o4) integrate(cos(3*x)/(5-4*cos(x)),x,0,10*%pi)=(5*%pi)/12
Wrong:
(%i5) integrate(f,x,0,10*%pi);
(%o5) 0