<p>Find \(\displaystyle\int_0^1 x^n(1-x)^n\,dx\) [JEE Main 2021]</p>
Step-by-Step Solution
Key Concept: Beta function: \int_0^1 xⁿ(1-x)ⁿ dx = B(n+1,n+1) = (n\!)^2/(2n+1)\!
<div class='solution'>
<p>$\int_0^1 x^n(1-x)^n\,dx = B(n+1,n+1)=\frac{\Gamma(n+1)\Gamma(n+1)}{\Gamma(2n+2)}=\frac{(n\!)^2}{(2n+1)\!}$</p>
</div>
Correct Answer: A