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Question

π6π3ln(sinx)dx12ln(34)ln2sin1exdx is

A
π6 ln(23)
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B
π6 ln(32)
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C
π3 ln(34)
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D
πln(43)
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Solution

The correct option is A π6 ln(23)
Let f(x)=ln(sinx)f1x=sin(ex)
π6π3ln(sinx)dx12ln(34)ln2sin1exdx
=π6π3f(x)dx12ln(34)ln2f1(x)dx
Let f1(x)=tx=f(t)dx=f(t) dt
π6π3f(x)dxπ3π6t.f(t)dt
=π6π3f(x)dx[[tf(t)]π3π6π3π6f(t)dt]
π6π3f(x)dx[tf(t)]π3π6π6π3f(t)dt
=π6ln2π6ln34=π6ln(23)

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