Integral of Arcsine Function

Theorem

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

for $x \in \left[{-1 \,.\,.\, 1}\right]$.

Proof

 * $\displaystyle \int \arcsin x \ \mathrm d x = \int 1 \cdot \arcsin x \ \mathrm d x$

Using:
 * Integration by Parts
 * Derivative of Arcsine Function
 * Integral of Constant

we obtain:


 * $\displaystyle \int \arcsin x \ \mathrm d x = x \arcsin x - \int \dfrac x {\sqrt {1 - x^2}} \ \mathrm dx$

Substitute: