Primitive of Function of Arcsine

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Theorem

$\ds \int \map F {\arcsin \frac x a} \rd x = a \int \map F u \cos u \rd u$

where $u = \arcsin \dfrac x a$.


Proof

First note that:

\(\ds u\) \(=\) \(\ds \arcsin \frac x a\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds a \sin u\) Definition of Real Arcsine


Then:

\(\ds u\) \(=\) \(\ds \arcsin \frac x a\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac 1 {\sqrt {a^2 - x^2} }\) Derivative of Arcsine Function: Corollary
\(\ds \leadsto \ \ \) \(\ds \int \map F {\arcsin \frac x a} \rd x\) \(=\) \(\ds \int \map F u \sqrt {a^2 - x^2} \rd u\) Primitive of Composite Function
\(\ds \) \(=\) \(\ds \int \map F u \sqrt {a^2 - a^2 \sin^2 u} \rd u\) Definition of $x$
\(\ds \) \(=\) \(\ds \int \map F u a \sqrt {1 - \sin^2 u} \rd u\)
\(\ds \) \(=\) \(\ds \int \map F u a \cos u \rd u\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds a \int \map F u \cos u \rd u\) Primitive of Constant Multiple of Function

$\blacksquare$


Also see


Sources

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