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Question Number 58362 by peter frank last updated on 22/Apr/19

$$\int \frac{\sin x}{\sin 3x}dx$$

Answered by mr W last updated on 22/Apr/19

$$=\int \frac{\sin x}{3\sin x-4{\sin }^{3}x}dx$$$$=\int \frac{1}{3-4{\sin }^{2}x}dx$$$$=\int \frac{1}{3-2(1-\cos 2x)}dx$$$$=\int \frac{1}{1+2\cos 2x}dx$$$$=\frac{1}{4}\int \frac{1}{\frac{1}{2}+\cos 2x}d2x$$$$=\frac{1}{4}\int \frac{1}{\frac{1}{2}+\cos t}dt$$$$=\frac{1}{2\sqrt{3}}\ln \mid \frac{\tan \frac{t}{2}+\sqrt{3}}{\tan \frac{t}{2}-\sqrt{3}}\mid +C$$$$=\frac{1}{2\sqrt{3}}\ln \mid \frac{\tan x+\sqrt{3}}{\tan x-\sqrt{3}}\mid +C$$

Commented by peter frank last updated on 23/Apr/19

$$thanks$$

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