# JEE Main & Advanced Mathematics Definite Integrals Gamma Function

Gamma Function

Category : JEE Main & Advanced

$\int_{0}^{\infty }{{{x}^{n-1}}}{{e}^{-x}}dx$, $n>0$ 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.

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