Question

integration DX/sin (x - a) sin (x-6)
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Answer 1

$\\ \color{violet} \large \underline{ \underline{ \bold{Solution :- }}}$

$\\ \: \: \: \sf \: \int \dfrac{dx}{ \sin(x - a) \sin(x - b) }$

$\\ \: \sf \: \: \: \: \: \: \:\: \qquad \: \: {\leadsto} \: \: \frac{1}{ \sin(b - a)} \: { \: \sf \: \int \dfrac{ \sin(b - a)}{ \sin(x - a) \sin(x - b) }} \: dx$

$\\ \: \sf \: \: \: \: \: \: \:\: \qquad \: \: {\leadsto} \: \: \frac{1}{ \sin(b - a)} \: { \: \sf \: \int \dfrac{ \sin(x - a) - \sin(x - b)}{ \sin(x - a) \sin(x - b) }} \: dx$

$\\ \: \sf \: \: \: \: \: \: \:\: \qquad \: \: {\leadsto} \: \: \frac{1}{ \sin(b - a)} \: { \: \sf \: \int \dfrac{ \sin(x - a) \times \cos(x - b) - \cos(x - a) \times \sin(x - b)}{ \sin(x - a) \sin(x - b) }} \: dx$

$\\ \: \sf \: \: \: \: \: \: \:\: \qquad \: \: {\leadsto} \: \: \frac{1}{ \sin(b - a)} \: \: \sf \: \int \:[ \cot(x - b) - \cot(x - a)]dx$

$\\ \: \sf \: \: \: \: \: \: \:\: \qquad \: \: {\leadsto} \: \: \frac{1}{ \sin(b - a)} \: \: \sf \: \:[ \log | \sin(x - b)| - \log | \sin(x - a)| ] + C$

$\\ \: \sf \: \: \: \: \: \: \:\: \qquad \: \: {\leadsto} \: \: \frac{1}{ \sin} (b - a)\: \: \sf \: \:[ \log | \sin(x - b)| - \log | \sin(x - a)| ] + C$

$\\ \: \sf \: \: \: \: \: \: \:\: \qquad \: \: {\leadsto} \: \: \cosec(b - a)\: \: \sf \: \:[ \log | \sin(x - b)| - \log | \sin(x - a)| ] + C$

$\\ \: \sf \: \: \: \: \: \: \:\: \qquad \: \: {\leadsto} \: \: \cosec(b - a)\: \: \sf \: \bigg[ \log \bigg| \dfrac{ \sin(x - b)}{\sin(x - a) } \bigg | \bigg] + C$

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