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Question
Evaluate the following:
`int sinx/(sin 3x) dx`
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Solution
Let I = `int sinx/(sin 3x) . dx`
= `int sinx/(3sinx - 4sin^3x) . dx`
= `int (sinx)/(sinx(3 - 4sin^2x)) . dx`
= `int (1)/(3 - 4sin^2x) . dx`
Dividing both numerator and denominator by cos2x, we get
I = `int (sec^2x)/(3sec^2x - 4tan^2x) . dx`
= `int (sec^2x)/(3(1 + tan^2x) - 4tan^2x) . dx`
= `int (sec^2x)/(3 - tan^2x) . dx`
Put tan x = t
∴ sec2x dx = dt
I = `int dt/(3-t^2)`
I = `int dt/((sqrt(3))^2 - t^2)`
= `int1/((sqrt3)^2 - t^2)dt`
= `(1)/(2sqrt(3)) log |(sqrt(3) + t)/(sqrt(3) - t)| + c`
= `(1)/(2sqrt(3)) log |(sqrt(3) + tanx)/(sqrt(3) - tanx)| + c`.
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