Integral of Arcsine Function

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Theorem

$\displaystyle \int \arcsin x \rd x = x \arcsin x + \sqrt {1 - x^2} + C$

for $x \in \closedint {-1} 1$.


Proof

\(\displaystyle \int \arcsin x \rd x\) \(=\) \(\displaystyle \int 1 \cdot \arcsin x \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle x \arcsin x - \int x \paren {\frac \rd {\rd x} \arcsin x} \rd x\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle x \arcsin x - \int \frac x {\sqrt {1 - x^2} } \rd x\) Derivative of Arcsine Function


Substitute:

\(\displaystyle u\) \(=\) \(\displaystyle 1 - x^2\)
\(\displaystyle \frac {\d u} {\d x}\) \(=\) \(\displaystyle -2 x\) differentiating with respect to $x$
\(\displaystyle \leadsto \ \ \) \(\displaystyle -\frac 1 2 \frac {\d u} {\d x}\) \(=\) \(\displaystyle x\)
\(\displaystyle \int \arcsin x \rd x\) \(=\) \(\displaystyle x \arcsin x - \int \frac {- \frac 1 2} {\sqrt u} \frac {\d u} {\d x} \rd x\)
\(\displaystyle \int \arcsin x \rd x\) \(=\) \(\displaystyle x \arcsin x + \frac 1 2 \int u^{-1/2} \rd u\) Integration by Substitution
\(\displaystyle \) \(=\) \(\displaystyle x \arcsin x + u^{1/2} + C\) Integral of Power
\(\displaystyle \) \(=\) \(\displaystyle x \arcsin x + \sqrt {1 - x^2} + C\) from $u = 1 - x^2$

$\blacksquare$


Sources