Question

integrate sin(x-a)/sin(x+a) dx

Answer 1

\[ \begin{align} \int\frac{\sin(x-a)}{\sin(x+a)}\,dx &= \int\frac{\sin\left((x+a)-2a\right)}{\sin(x+a)}\, dx \\~\\ &\text{use identity:}\sin(x+y)=\sin x\cos y- \cos x \sin y \\ ~ \\ &= \int\frac{\sin(x+a)\cos(2a)-\cos(x+a)\sin(2a)}{\sin(x+a)} \, dx \\ ~ \\ &=\int\frac{\sin(x+a)\cos(2a)}{\sin(x+a)} \, dx-\int\frac{\cos(x+a)\sin(2a)}{\sin(x+a)} \, dx \\ ~ \\ &=\int \cos(2a) \, dx -\int\cot(x+a)\sin(2a) \, dx \\ ~ \\ &\text{notice that } \cos(2a) \text{ and } \sin(2a) \text{ are constants with respect to } x \\ ~ \\& =\cos(2a) \cdot x-\sin(2a)\left( \ln|\sin (x+a)| \right)+C \\
&=x\cos(2a)-\sin(2a)\ln|\sin(x+a)|+C\end{align} \]