Primitive of Reciprocal of Root of 1 minus x squared/Arcsine Form

Theorem

$\ds \int \frac {\d x} {\sqrt {1 - x^2} } = \arcsin x + C$

where $C$ is an arbitrary constant.

Proof 1

From Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$: Arcsine Form:

$\ds \int \frac {\d x} {\sqrt {a^2 - x^2} } = \arcsin \frac x a + C$

The result follows by setting $a = 1$.

$\blacksquare$

Proof 2

\(\ds \map {\dfrac \d {\d x} } {\arcsin x}\)

\(=\)

\(\ds \dfrac 1 {\sqrt {1 - x^2} }\)

Derivative of Arcsine Function

\(\ds \leadsto \ \ \)

\(\ds \int \dfrac {\d x} {\sqrt {1 - x^2} }\)

\(=\)

\(\ds \arcsin x + C\)

Definition of Primitive (Calculus)

$\blacksquare$