Beta
Relation between Beta and Gamma functions: \(\beta(m, n) = \dfrac{\Gamma(m) \Gamma(n)}{\Gamma(m + n)}\)
\(\beta(m, n) = \int_0^1 x^{m-1} (1 - x)^{n-1} \, dx \quad (m, n > 0)\)
Symmetric property: \(\beta(m, n) = \beta(n, m)\)
\(\beta(m, n) = 2 \int_0^{\pi/2} (\sin \theta)^{2m-1} (\cos \theta)^{2n-1} \, d\theta\)

1) Prove that, \(\Gamma(1 / 2) = \sqrt{\pi}\)
2) Evaluate \(B(3, 2)\)
3) Show that \(B(m, n) = \dfrac{\Gamma(m)\Gamma(n)}{\Gamma(m + n)}\)
4) Find the value of \(B\left(\dfrac{1}{2}, \dfrac{1}{2}\right)\)
5) Prove that \(B(1, n) = \dfrac{1}{n}\)
6) Evaluate \(B(2, 3)\) using Gamma functions
7) Express \(B(m, n)\) as an integral involving sine and cosine:
\(B(m, n) = 2 \int_0^{\pi/2} (\sin \theta)^{2m-1} (\cos \theta)^{2n-1} \, d\theta\)
8) Evaluate \(\int_0^{\pi/2} \sin^4\theta \cos^2\theta \, d\theta\) using Beta function
9) Evaluate \(\int_0^{\pi/2} \sin^5\theta \, d\theta\)
10) Show that \(\int_0^{\pi/2} \sin^{m-1}\theta \cos^{n-1}\theta \, d\theta = \dfrac{1}{2}B\left(\dfrac{m}{2}, \dfrac{n}{2}\right)\)
11) If \(\Gamma(m)\Gamma(1 - m) = \dfrac{\pi}{\sin(\pi m)}\), find the value of \(\Gamma\left(\dfrac{3}{4}\right)\Gamma\left(\dfrac{1}{4}\right)\)
12) Evaluate \(\Gamma(5)\)
13) Prove that \(\Gamma(n + 1) = n\Gamma(n)\) for all positive integers \(n\)
14) Show that \(\Gamma(n + 1) = n!\)
15) Find the value of \(\Gamma\left(\dfrac{3}{2}\right)\)
16) Prove that \(\Gamma\left(\dfrac{5}{2}\right) = \dfrac{3}{4}\sqrt{\pi}\)

Differential & Integral Calculus