If ∫0∞sinxxdx=π2 then ∫0∞sin3xxdx=

I=∫0∞3sinx-sin3x4xdx=34×π2-14∫0∞sin3xxdx =3π8-14∫0∞sintt3dt3=3π8-14∫0∞sinxxdx=3π8-14×π2=π4

If ∫0∞sinxxdx=π2 then ∫0∞sin3xxdx=

I=∫0∞3sinx-sin3x4xdx=34×π2-14∫0∞sin3xxdx =3π8-14∫0∞sintt3dt3=3π8-14∫0∞sinxxdx=3π8-14×π2=π4

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$If \int_{0}^{\infty} \frac{sinx}{x} d x = \frac{π}{2} then \int_{0}^{\infty} \frac{\sin^{3} x}{x} d x =$

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3 π 2

π 6

π 2

π 4

answer is B.

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Detailed Solution

$I = \int_{0}^{\infty} \frac{3 \sin x - \sin 3 x}{4 x} d x = \frac{3}{4} \times \frac{π}{2} - \frac{1}{4} \int_{0}^{\infty} \frac{\sin 3 x}{x} d x = \frac{3 π}{8} - \frac{1}{4} \int_{0}^{\infty} \frac{\sin t}{\frac{t}{3}} (\frac{d t}{3}) = \frac{3 π}{8} - \frac{1}{4} \int_{0}^{\infty} \frac{\sin x}{x} d x = \frac{3 π}{8} - \frac{1}{4} \times \frac{π}{2} = \frac{π}{4}$