What is the integral of #|sin(x)|#?

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Answer 1 · 2014-12-22T05:03:38

The function #|sin(x)|# is defined as follows:

#|sin(x)| = sin(x), if sin(x) geq 0#
#|sin(x)| = -sin(x), if sin(x) < 0#

So, the integral is defined as:

#int |sin(x)| dx = int sin(x) dx, if sin(x) geq 0#
#int |sin(x)| dx = int -sin(x) dx, if sin(x) < 0#

Since the integral is linear:

#int -sin(x) dx = - int sin(x) dx#

And we have:

#int |sin(x)| dx = -cos(x) + C, if sin(x) geq 0#
#int |sin(x)| dx = cos(x) + C, if sin(x) < 0#

or, for #n in ZZ#,

#int |sin(x)| dx = -cos(x) + C, if x in [2npi, (2n+1)pi]#
#int |sin(x)| dx = cos(x) + C, if x in ((2n-1)pi, 2npi)#