int_0^{pi /2} sin^4 x cos^6 x , dx ની કિંમત શ | vedclass

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(B) વોલિસના સૂત્રનો ઉપયોગ કરતા,$I = \int_0^{\pi /2} \sin^m x \cos^n x \, dx = \frac{\Gamma(\frac{m+1}{2}) \Gamma(\frac{n+1}{2})}{2 \Gamma(\frac{m+n+2}

Vedclass · Accepted Answer

$\frac{3\pi}{512}$

Vedclass · Answer

(B) વોલિસના સૂત્રનો ઉપયોગ કરતા,$I = \int_0^{\pi /2} \sin^m x \cos^n x \, dx = \frac{\Gamma(\frac{m+1}{2}) \Gamma(\frac{n+1}{2})}{2 \Gamma(\frac{m+n+2}{2})}$.અહીં,$m = 4$ અને $n = 6$ છે.$I = \frac{\Gamma(\frac{4+1}{2}) \Gamma(\frac{6+1}{2})}{2 \Gamma(\frac{4+6+2}{2})} = \frac{\Gamma(5/2) \Gamma(7/2)}{2 \Gamma(6)}$.$\Gamma(n+1) = n\Gamma(n)$ અને $\Gamma(1/2) = \sqrt{\pi}$ નો ઉપયોગ કરતા:$\Gamma(5/2) = \frac{3}{2} \cdot \frac{1}{2} \cdot \sqrt{\pi} = \frac{3\sqrt{\pi}}{4}$.$\Gamma(7/2) = \frac{5}{2} \cdot \frac{3}{2} \cdot \frac{1}{2} \cdot \sqrt{\pi} = \frac{15\sqrt{\pi}}{8}$.$\Gamma(6) = 5! = 120$.$I = \frac{(\frac{3\sqrt{\pi}}{4}) (\frac{15\sqrt{\pi}}{8})}{2 \cdot 120} = \frac{45\pi}{32 \cdot 240} = \frac{45\pi}{7680} = \frac{3\pi}{512}$.

$\int_0^{\pi /2} \sin^4 x \cos^6 x \, dx$ ની કિંમત શોધો.

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