Evaluate:`int(sinx)/(sin4x)dx`

1.	Evaluate:`int(sinx)/(sin4x)dx`
Answer» `I=int(sinx)/(sin4x)dx=int(sinx)/(2sin2x cos2x)dx` `=int(sinx)/(4sinx cos x cos2x)dx` `=(1)/(4)int(1)/(cosx cos2x)dx=(1)/(4)int(cosx)/(cos^(2)x cos2x)dx` `=(1)/(4)int(cosx)/((1-sin^(2)x)(1-2sin^(2)x))dx` Putting `sinx =t` and `cosx dx=dt,` we get `I=(1)/(4)int(dt)/((1-t^(2))(1-2t^(2)))` `=(1)/(4)int[(2)/(1-2t^(2))-(1)/(1-t^(2))]dt` `= -(1)/(4)int(1)/(1-t^(2))dt+(2)/(4)int(1)/(1-(sqrt(2)t)^(2))dt` `= -(1)/(4)xx(1)/(2)log\|(1+t)/(1-t)\|+(1)/(2).(1)/(2sqrt(2))log\|(1+sqrt(2)t)/(1-sqrt(2)t)\| +C` `= -(1)/(8)log\|(1+sinx)/(1-sinx)\|+(1)/(4sqrt(2))log\|(1+sqrt(2)sinx)/(1-sqrt(2)sinx)\| +C`

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