JEE Main & Advanced Mathematics Definite Integration Gamma Function

\[\int_{0}^{\infty }{{{x}^{n-1}}}{{e}^{-x}}dx\], \[n>0\] is called Gamma function and denoted by \[\Gamma n\]. If \[m\] and \[n\] are non-negative integers, then

\[\int_{0}^{\pi /2}{{{\sin }^{m}}x{{\cos }^{n}}xdx}=\frac{\Gamma \left( \frac{m+1}{2} \right)\,\Gamma \left( \frac{n+1}{2} \right)}{2\Gamma \left( \frac{m+n+2}{2} \right)}\]

where \[\Gamma (n)\] is called gamma function which satisfy the following properties \[\Gamma (n+1)=n\Gamma (n)=n!\]i.e.,\[\Gamma \,(1)=1\], \[\Gamma (1/2)=\sqrt{\pi }\]

In place of gamma function, we can also use the following formula \[\int_{0}^{\pi /2}{{{\sin }^{m}}x{{\cos }^{n}}xdx}\]

=\[\frac{(m-1)(m-3).....(2\text{ or }1)(n-1)(n-3).....(2\text{ or 1)}}{(m+n)(m+n-2)....(2\text{ or }1)}\]

It is important to note that we multiply by \[(\pi /2);\] when both \[m\] and \[n\] are even.