int_{-pi / 2}^{pi / 2} sin ^4 x cos ^6 x , dx | vedclass

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Question

(B) અહીં વિધેય $f(x) = \sin^4 x \cos^6 x$ એ યુગ્મ વિધેય હોવાથી: $\int_{-\pi / 2}^{\pi / 2} \sin ^4 x \cos ^6 x \, dx = 2 \int_0^{\pi / 2} \sin ^4 x \c

Vedclass · Accepted Answer

$\frac{3 \pi}{256}$

Vedclass · Answer

(B) અહીં વિધેય $f(x) = \sin^4 x \cos^6 x$ એ યુગ્મ વિધેય હોવાથી: $\int_{-\pi / 2}^{\pi / 2} \sin ^4 x \cos ^6 x \, dx = 2 \int_0^{\pi / 2} \sin ^4 x \cos ^6 x \, dx$ વોલિસના સૂત્ર $\int_0^{\pi/2} \sin^m x \cos^n x \, dx = \frac{\Gamma(\frac{m+1}{2}) \Gamma(\frac{n+1}{2})}{2 \Gamma(\frac{m+n+2}{2})}$ નો ઉપયોગ કરતા: $I = 2 	imes \frac{\Gamma(\frac{4+1}{2}) \Gamma(\frac{6+1}{2})}{2 \Gamma(\frac{4+6+2}{2})} = \frac{\Gamma(\frac{5}{2}) \Gamma(\frac{7}{2})}{\Gamma(6)}$ $\Gamma(n+1) = n\Gamma(n)$ અને $\Gamma(\frac{1}{2}) = \sqrt{\pi}$ નો ઉપયોગ કરતા: $\Gamma(\frac{5}{2}) = \frac{3}{2} \cdot \frac{1}{2} \sqrt{\pi} = \frac{3}{4} \sqrt{\pi}$ $\Gamma(\frac{7}{2}) = \frac{5}{2} \cdot \frac{3}{2} \cdot \frac{1}{2} \sqrt{\pi} = \frac{15}{8} \sqrt{\pi}$ $\Gamma(6) = 5! = 120$ $I = \frac{(\frac{3}{4} \sqrt{\pi}) (\frac{15}{8} \sqrt{\pi})}{120} = \frac{45 \pi}{32 	imes 120} = \frac{3 \pi}{256}$

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

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