Question

the value of beta (1/2,1/2)=​

Answer 1

$\displaystyle \sf{ \beta \bigg( \frac{1}{2} , \frac{1}{2} \bigg) = \pi}$

Given :

$\displaystyle \sf{ \beta \bigg( \frac{1}{2} , \frac{1}{2} \bigg)}$

To find :

The value

Solution :

Step 1 of 2 :

Define Beta function

We know that for m , n > 0

$\displaystyle \sf \beta (m,n) =2 \int\limits_{0}^{ \frac{\pi}{2} } {sin}^{2m - 1} x \: {cos}^{2n - 1} x \: \, dx$

Step 2 of 2 :

Find the value

$\displaystyle \sf{ We \: put \: m = \frac{1}{2} \: \: and \: \: n = \frac{1}{2} }$

Thus we get

$\displaystyle \sf{ \beta \bigg( \frac{1}{2} , \frac{1}{2} \bigg)}$

$\displaystyle \sf = 2\int\limits_{0}^{ \frac{\pi}{2} } \: \, dx$

$\displaystyle \sf = 2 \: x \bigg| _{0}^{ \frac{\pi}{2} }$

$\displaystyle \sf = 2 \bigg[ \frac{\pi}{2} - 0\bigg]$

$\displaystyle \sf = 2 \times \frac{\pi}{2}$

$\displaystyle \sf =\pi$

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