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Question
\[\int\frac{\sin 2x}{\sin 5x \sin 3x} dx\]
Sum
Solution
\[\int\frac{\sin 2x}{\sin 5x \sin 3x}dx\]
\[ = \int\frac{\sin \left( 5x - 3x \right)}{\sin 5x \sin 3x}dx\]
\[ = \int\frac{\sin 5x \cos 3x - \cos 5x \sin 3x}{\sin 5x \sin 3x}dx\]
\[ = \int \frac{\sin 5x \cos 3x}{\sin 5x \sin 3x} - \frac{\cos 5x \sin 3x}{\sin 5x \sin 3x}dx\]
\[ = \int\left[ \cot 3x - \cot 5x \right] dx\]
\[ = ∫ cot\ 3x\ dx - \int\cot\ 5x\ dx\]
\[ = \frac{1}{3} \text{ln} \left| \sin 3x \right| - \frac{1}{5} \text{ln }\left| \sin 5x \right| + C\]
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