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

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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
\(\ds \) \(=\) \(\ds \frac \pi 2\)

$\blacksquare$


Sources

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