Question

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

Answer 1

$\int \frac{sinx}{sin3x}dx$

$=\int \frac{sinx}{3sinx-4si{n}^{3}x}dx$

$=\int \frac{dx}{3-4si{n}^{2}x}$

$=\int \frac{dx}{\frac{3-4(1-cos2x)}{2}}$

$=\int \frac{dx}{3-2(1-cos2x)}$

$=\int \frac{dx}{3-2+2cox2x}$

$=\int \frac{dx}{1+2cos2x}$

$=\int \frac{dx}{1+\frac{2(1-t^2)}{1+t^2}}$

t = tanx

$dt=se{c}^{2}xdx\Rightarrow dx=\frac{dt}{1+t^2}$

$=\int \frac{dt}{1+t^2+2-2t^2}$

$=\int \frac{dt}{3-t^2}$

$\int \frac{sinx}{sin3x}dx=\frac{1}{2\sqrt{3}}ln\left(\left|\frac{\sqrt{3}+tanx}{\sqrt{3}-tanx}\right|\right)+c$