# Definite Integral from 0 to a of Reciprocal of Root of a Squared minus x Squared

## Theorem

$\ds \int_0^a \dfrac {\d x} {\sqrt {a^2 - x^2} } = \frac \pi 2$

for $a > 0$.

## Proof 1

 $\ds \int_0^a \dfrac {\d x} {\sqrt {a^2 - x^2} }$ $=$ $\ds \int_0^{\mathop \to a} \dfrac {\d x} {\sqrt {a^2 - x^2} }$ as $\dfrac 1 {\sqrt {a^2 - x^2} }$ does not exist for $x = a$ $\ds$ $=$ $\ds \lim_{\gamma \mathop \to a} \int_0^\gamma \dfrac {\d x} {\sqrt {a^2 - x^2} }$ Definition of Improper Integral on Open Above Interval $\ds$ $=$ $\ds \lim_{\gamma \mathop \to a} \intlimits {\arcsin \frac x a} 0 \gamma$ Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$ $\ds$ $=$ $\ds \lim_{\gamma \mathop \to a} \paren {\arcsin \frac \gamma a - \arcsin 0}$ $\ds$ $=$ $\ds \lim_{\gamma \mathop \to a} \arcsin \frac \gamma a$ Arcsine of Zero is Zero $\ds$ $=$ $\ds \arcsin \frac a a$ $\ds$ $=$ $\ds \arcsin 1$ $\ds$ $=$ $\ds \frac \pi 2$ Arcsine of One is Half Pi

$\blacksquare$

## Proof 2

 $\ds \int_0^a \frac {\d x} {\sqrt {a^2 - x^2} }$ $=$ $\ds \frac {a^{1 - \frac 2 2} \map \Gamma {\frac 1 2} \map \Gamma {-\frac 1 2 + 1} } {2 \map \Gamma {\frac 1 2 - \frac 1 2 + 1} }$ Definite Integral from 0 to a of $x^m \paren {a^n - x^n}^p$ $\ds$ $=$ $\ds \frac 1 {2 \times 0!} \paren {\map \Gamma {\frac 1 2} }^2$ Gamma Function Extends Factorial $\ds$ $=$ $\ds \frac 1 2 \paren {\sqrt \pi}^2$ Gamma Function of One Half $\ds$ $=$ $\ds \frac \pi 2$

$\blacksquare$