Integrate the function x (x-a) - Vidyadip

Question

Integrate the function $\frac{\sin x}{\sin (x-a)}$

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Answer

Given Integrand is: $\frac{\sin x}{\sin (x-a)}$
Let $\mathrm{I}=\int\frac{\sin \mathrm{x}}{\sin (\mathrm{x}-\mathrm{a})}$
Let x - a = t $\Rightarrow$ x = t + a $\Rightarrow$ dx = dt
$\Rightarrow \int \frac{\sin x}{\sin (x-a)} d x=\int \frac{\sin (t+a)}{\sin (t)} d t$
As, {sin (A + B) = sin A cos B + cos A sin B}
$\Rightarrow \int \frac{\sin x}{\sin (x-a)} d x$ = $\int \frac{\sin t \cos a+\cos t \sin a}{\sin (t)} d t$
$=\int \frac{\sin t \cos a}{\sin t}+\frac{\cos t \sin a}{\sin t} d t$
= $\int(\cos a+\cot t \sin a) d t$
= $\int(\cos a) d t+\int(\cot t \sin a) d t$
= $\cos a \int 1 . \mathrm{dt}+\sin \mathrm{a} . \int(\cot t) \mathrm{d} \mathrm{t}$
= $(\cos a) \cdot(x-a)+\sin a \cdot \log |\sin (x-a)|+c$
= $\sin a \cdot \log |\sin (x-a)|+x \cdot \cos a-a \cdot \cos a+c$
= $\sin a \cdot \log |\sin (x-a)|+x \cdot \cos a+c_{1}$

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