Hi. For an integral like this, for example:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\int {\sqrt {1 - \cos ^2 x} dx}

[/tex]

The most obvious way of solving would be to make use of the pythagorean identity, to get:

[tex]

\int {\sqrt {\sin ^2 x} dx}

[/tex]

Now, ive been taught to simply evaluate it like this:

[tex]

\int {\sqrt {\sin ^2 x} dx} = \int {\sin xdx = - \cos x + C}

[/tex]

I was wondering though, is that technically correct? Squaring something, then taking the square root of it is equivalent of taking the absolute value of it, so would this be more correct:

[tex]

\int {\sqrt {\sin ^2 x} dx} = \int {\left| {\sin x} \right|dx}

[/tex]

If so, would i just give two different solutions, each defined over a different domain, like a piecewise function?

[tex]

\int {\left| {\sin x} \right|dx = \left\{ {\begin{array}{*{20}c}

{ - \cos x + C} & {\sin x \ge 0} \\

{\cos x + C} & {\sin x < 0} \\

\end{array}} \right.}

[/tex]

Is that a more correct way of doing it?

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# Absolute Value Integrals

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