Primitives of Functions of Inverse Trigonometric Functions

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Theorem

Primitive of Function of Arcsine

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

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


Primitive of Function of Arccosine

$\ds \int \map F {\arccos \frac x a} \rd x = -a \int \map F u \sin u \rd u$

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


Primitive of Function of Arctangent

$\ds \int \map F {\arctan \frac x a} \rd x = a \int \map F u \sec^2 u \rd u$

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


Primitive of Function of Arccotangent

$\ds \int \map F {\arccot \frac x a} \rd x = -a \int \map F u \csc^2 u \rd u$

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


Primitive of Function of Arcsecant

$\ds \int \map F {\arcsec \frac x a} \rd x = a \int \map F u \sec u \tan u \rd u$

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


Primitive of Function of Arccosecant

$\ds \int \map F {\arccsc \frac x a} \rd x = -a \int \map F u \size {\csc u} \cot u \rd u$

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