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Prove-0-sin-n-x-x-m-dx-1-m-0-D-m-1-sin-n-x-x-dx-n-m-Odd-Number-




Question Number 159931 by qaz last updated on 22/Nov/21
Prove ::    ∫_0 ^∞ ((sin^n x)/x^m )dx=(1/(Γ(m)))∫_0 ^∞ ((D^(m−1) sin^n x)/x)dx  n+m∈Odd Number.
$$\mathrm{Prove}\:::\:\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}^{\mathrm{n}} \mathrm{x}}{\mathrm{x}^{\mathrm{m}} }\mathrm{dx}=\frac{\mathrm{1}}{\Gamma\left(\mathrm{m}\right)}\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{D}^{\mathrm{m}−\mathrm{1}} \mathrm{sin}^{\mathrm{n}} \mathrm{x}}{\mathrm{x}}\mathrm{dx} \\ $$$$\mathrm{n}+\mathrm{m}\in\mathrm{Odd}\:\mathrm{Number}. \\ $$

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