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

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Answer 1 · 2014-12-22T05:03:38

The function $| sin (x) |$ $|sin(x)|$ 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 $n in ZZ$ ,

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

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