Question

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

Answer 1

Special Integrals -

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

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

$\Rightarrow \int \frac{dx}{3-4si{n}^{2}x}$ $sin3\theta =3sin\theta -4si{n}^3\theta$

$\Rightarrow \int \frac{dx}{3-4\frac{1-cos2x}{2}}$

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

$\Rightarrow \int \frac{dx}{3-2+2cos2x}$ $cos2\theta =\frac{1-ta{n}^2\theta }{1+ta{n}^2\theta }$

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

$\Rightarrow \int \frac{dx}{1+\frac{2(1+t^2)}{1+t^2}}$ (where $t=tanxdt=se{c}^{2}xdxdx=\frac{dt}{1+t^2})$

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

$\Rightarrow \frac{dt}{3-t^2}$

$\Rightarrow \int \frac{1}{2.\sqrt{3}}ln\left(|\frac{\sqrt{3}+tanx}{\sqrt{3}-tanx|}\right)+c$

using $\int \frac{1}{x^2-a^2}=\frac{1}{2a}ln|\frac{a-x}{a+x}|$