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Question: If I = \(\int_{1/3}^{3}{\frac{1}{x}\sin\left( \frac{1}{x} - x \right)}\)dx, then I equals :...

If I = ∫1/331xsin⁡(1x−x)\int_{1/3}^{3}{\frac{1}{x}\sin\left( \frac{1}{x} - x \right)}dx, then I equals :

A

3\sqrt{3}/2

B

p +3\sqrt{3}/2

C

0

D

None of these

Answer

0

Explanation

Solution

Put x = 1t\frac{1}{t}

I = ∫31/3tsin⁡(t−1t)(−1t2)dt\int_{3}^{1/3}{t\sin\left( t - \frac{1}{t} \right)}\left( - \frac{1}{t^{2}} \right)dt

= ∫31/3sin⁡(1t−t)1tdt\int_{3}^{1/3}{\sin\left( \frac{1}{t} - t \right)}\frac{1}{t}dt

= – I Ž I = 0