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Question

The value of sinxsin3xdx is
(where C is integration constant)

A
13ln3tanx3+tanx+C
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B
123ln23tanx23+tanx+C
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C
123ln3+tanx3tanx+C
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D
13ln23+tanx23tanx+C
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Solution

The correct option is C 123ln3+tanx3tanx+C

I=sinxsin3xdx =sinx3sinx4sin3xdx =134sin2xdx(sinx0)
Dividing numerator and denominator by cos2x,
I=sec2x3sec2x4tan2xdx
Putting tanx=t
sec2xdx=dt
Now,
I=dt3(1+t2)4t2 =dt3t2 =1(3)2t2dt =123ln3+t3t+C[1a2x2dx=12alna+xax+C] =123ln3+tanx3tanx+C


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