Question Number 54765 by peter frank last updated on 10/Feb/19
Answered by tanmay.chaudhury50@gmail.com last updated on 10/Feb/19
$$\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{2}×\mathrm{24}−\mathrm{1}} {xcos}^{\mathrm{2}×\mathrm{27}−\mathrm{1}} {xdx} \\ $$$${formula}\:\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{2}{p}−\mathrm{1}} {xcos}^{\mathrm{2}{q}−\mathrm{1}} {xdx} \\ $$$$=\frac{\lceil\left({p}\right)\lceil\left({q}\right)}{\lceil\left({p}+{q}\right)} \\ $$$${so}\:{ans}\:{is}=\frac{\mathrm{1}}{\mathrm{2}}×\frac{\lceil\left(\mathrm{24}\right)×\lceil\left(\mathrm{27}\right)}{\lceil\left(\mathrm{51}\right)}=\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{23}!\mathrm{26}!}{\mathrm{50}!} \\ $$$${firmula}\:{gamma}\:{function}\rightarrow\lceil\left({n}+\mathrm{1}\right)={n}!\left({factirial}\right) \\ $$$$ \\ $$$$ \\ $$