Finding the definite integral of the cosine modulus?

Question

Mathematics

Hello, in the subject, unfortunately, not too strong. Dug out the following formula:
∫|cos x| dx = sinx*sgn(cos x) + C Checked, seems to be correct. Now I want to find a definite integral in terms of 0 ; 10pi and substitute in the Newton-Leibniz formula (the only condition for this is the continuity of the function, which seems to be true for the cosine modulus) and it turns out that it is equal to sin 10pi * sgn (cos 10pi) + C - sin 10pi * sgn (cos 10pi ) - C , which is zero, although it is obvious that the area under the graph of this function is not zero, but I just can’t figure out what I’m doing wrong.

Answer 1

The problem is that this is your formula - it only works on a segment where sinx * sgn (cos x) is continuous. And it has discontinuities at the points where the cosine changes sign. You apparently picked it up by typing. Indeed, if we take its derivative at each point, we get |cos x|. But this derivative cannot be taken at the points where the cosine changes sign, because there are discontinuities. It is not an antiderivative, if only because the antiderivative must be continuous. Therefore, the formula for a definite integral does not work here.

Answer 2

if you have a module, then just break your interval into segments on which the sign of the function does not change. Integrate the positive on the segments as usual, the negative - with a minus sign. Seems obvious.