Primitive of Arcsine of a x

Theorem

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

Proof

\(\ds \int \arcsin x \rd x\)

\(=\)

\(\ds x \arcsin x + \sqrt {1 - x^2} + C\)

Primitive of $\arcsin x$

\(\ds \leadsto \ \ \)

\(\ds \int \arcsin a x \rd x\)

\(=\)

\(\ds \dfrac 1 a \paren {\paren {a x} \arcsin a x + \sqrt {1 - \paren {a x}^2} } + C\)

Primitive of Function of Constant Multiple

\(\ds \)

\(=\)

\(\ds x \arcsin a x + \dfrac 1 a \sqrt {1 - a^2 x^2} + C\)

simplifying

$\blacksquare$

Also see

• Primitive of $\arccos a x$
• Primitive of $\arctan a x$

Sources

• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $96$.