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∫[0, ∞] sin^3x/x^3 dx
= -1/2*∫[0, ∞] sin^3x d(1/x^2)
= - sin^3x/2x^2 |[0, ∞] +1/2 ∫[0, ∞] 3sin^2xcosx/x^2 dx
= -3/2*∫[0, ∞] sin^2xcosx d(1/x)
= -3/2*sin^2xcosx /x |[0, ∞] + 3/2*∫[0, ∞] [2sinxcos^2x-sin^3x]/x dx
= 3/2*∫[0, ∞] [(3sin3x - sinx)/4]/x dx
= 3/8*∫[0, ∞] (3sin3x /x - sinx/x )dx
= 3π/8
= -1/2*∫[0, ∞] sin^3x d(1/x^2)
= - sin^3x/2x^2 |[0, ∞] +1/2 ∫[0, ∞] 3sin^2xcosx/x^2 dx
= -3/2*∫[0, ∞] sin^2xcosx d(1/x)
= -3/2*sin^2xcosx /x |[0, ∞] + 3/2*∫[0, ∞] [2sinxcos^2x-sin^3x]/x dx
= 3/2*∫[0, ∞] [(3sin3x - sinx)/4]/x dx
= 3/8*∫[0, ∞] (3sin3x /x - sinx/x )dx
= 3π/8
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