∫dxsin(x−a)sin(x−b)
1sin(b−a)∫sin[x−a−(x−b)]dxsin(x−a)sin(x−b) [∵sin[x−a−(x−b)]=sin[b−a]]
using [sin(A−B)=sinAcosB−cosAsinB]
1sin(b−a)∫sin(x−a)cos(x−b)−cos(x−a)sin(x−b)sin(x−a)sin(x−b)dx
1sin(b−a)∫sin(x−a)cos(x−b)sin(x−a)sin(x−b)−cos(x−a)sin(x−b)sin(x−a)sin(x−b)dx
1sin(b−a)∫[cot(x−b)−cot(x−a)]dx
1sin(b−a)[ln|sin(x−b)|−ln|sin(x−a)|]+C [∵∫cotx=ln|sinx|]
1sin(b−a)ln∣∣sin(x−b)sin(x−a)∣∣+C [∵lna−lnb=lnab]
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