Determine the indefinite integral $\int \frac{\sin x}{\sin 5x}dx$

Question

Find the value of the given integral

$$\int \frac{\sin x}{\sin 5x}dx$$

Answer 1

HINT:

$$\sin(5x) =5\sin(x)-20\sin^3(x)+16\sin^5(x)$$

$$\frac{\sin x}{\sin5x}=\frac1{5-20\sin^2x+16\sin^4x}$$

$$=\frac{\csc^4x}{5\csc^4x-20\csc^2x+16}$$

$$=\frac{(1+\cot^2x)\csc^2x}{5(1+\cot^2x)^2-20(1+\cot^2x)+16}$$

Set $\cot x=u$