Integrate a piecewise function (Second fundamental theorem of calculus)

I searched the forum but was not able to find a solution haw to integrate piecewise functions. The threads I found weren't clear either.

How can I integrate the following function for example?

F(x) = inntegral from 0 to x of f(t) dt

f(x) = x for 0 <= x <= 1
f(x) = x - 1 for 1 < x <= 2

Or is that even possible? Thank you!

[James Tursa]

What have you done so far? Why can't you just integrate each piece separately and combine appropriately?

Answer 1

[Karan Gill]

This is the same question as <http://www.mathworks.com/matlabcentral/newsreader/view_thread/347103>

Start with the piecewise function: https://www.mathworks.com/help/symbolic/piecewise.html

>> syms x
>> f(x) = piecewise(0<=x<=1, x, 1<x<2, x-1)
f(x) =
piecewise(x in Dom::Interval([0], [1]), x, x in Dom::Interval(1, 2), x - 1)

Get the integral using int.

>> syms F(x)
>> F(x) = int(f(x),x,0,x)
F(x) =
intlib::intOverSet(piecewise(x in Dom::Interval([0], [1]), x, x in Dom::Interval(1, 2), x - 1), x, [0, x])

Those output constructs are ugly but it's still better than going into MuPAD. Now you can do things like evaluate F(x).

>> F(1)
ans =
1/2

Answer 2

[Jan]

You have to integrate it in pieces. Whenever you try to integrate it in one piece, the discontinuity will conflict with the design of the integrators.

This soultion sounds trivial. Perhaps in the other threads you are talking of the users hesitated to post it. But sometimes trivial solutions are not obvious, when you are deeply involved in the problem.

Or I've overseen a detail. Then please explain this.