Integrating Absolute Value in Trigonometric Integrals

[NanakiXIII]

Homework Statement

I need to solve the integral

$$\int \frac{dx}{\sin{x} |\sin{x}|}.$$

Homework Equations

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The Attempt at a Solution

I know it's possible, the solution is $$-\frac{\cos{x} |\sin{x}|}{\sin^2{x}}$$, and I probably need to write the absolute as the square root of a square. I also probably need a substitution to simplify. I'm not having any luck finding such a substitution, however.

 

[ideasrule]

Integrate it for sinx>0, then for sinx<0. Compare.

 

[NanakiXIII]

That worked perfectly. Thanks. I suppose that's the usual way to tackle an integral over an absolute value?