# Integral of Arcsine Function

 It has been suggested that this page or section be merged into Primitive of Arcsine of x over a. (Discuss)

## Theorem

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

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

## Proof

 $\ds \int \arcsin x \rd x$ $=$ $\ds \int 1 \cdot \arcsin x \rd x$ $\ds$ $=$ $\ds x \arcsin x - \int x \paren {\frac \rd {\rd x} \arcsin x} \rd x$ Integration by Parts $\ds$ $=$ $\ds x \arcsin x - \int \frac x {\sqrt {1 - x^2} } \rd x$ Derivative of Arcsine Function
 $\ds u$ $=$ $\ds 1 - x^2$ $\ds \frac {\d u} {\d x}$ $=$ $\ds -2 x$ differentiating with respect to $x$ $\ds \leadsto \ \$ $\ds -\frac 1 2 \frac {\d u} {\d x}$ $=$ $\ds x$ $\ds \int \arcsin x \rd x$ $=$ $\ds x \arcsin x - \int \frac {- \frac 1 2} {\sqrt u} \frac {\d u} {\d x} \rd x$ $\ds \int \arcsin x \rd x$ $=$ $\ds x \arcsin x + \frac 1 2 \int u^{-1/2} \rd u$ Integration by Substitution $\ds$ $=$ $\ds x \arcsin x + u^{1/2} + C$ Integral of Power $\ds$ $=$ $\ds x \arcsin x + \sqrt {1 - x^2} + C$ from $u = 1 - x^2$

$\blacksquare$