Primitive of Arccosine Function

Theorem

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

Corollary

$\ds \int \arccos \frac x a \rd x = x \arccos \frac x a - \sqrt {a^2 - x^2} + C$

Proof 1

Let:

$\ds u$

$=$

$\ds \arccos x$

$\text {(1)}: \quad$

$\ds \leadsto \ \ $

$\ds \cos u$

$=$

$\ds x$

Definition of Real Arccosine

$\text {(2)}: \quad$

$\ds \leadsto \ \ $

$\ds \cos u$

$=$

$\ds \sqrt {1 - x^2}$

Sum of Squares of Sine and Cosine

Then:

$\ds \int \arccos x \rd x$

$=$

$\ds -\int u \sin u \rd u$

Primitive of Function of Arccosine

$\ds $

$=$

$\ds -\paren {\sin u - u \cos u} + C$

Primitive of $x \sin a x$, setting $a := 1$

$\ds $

$=$

$\ds -\paren {\sin u - u x} + C$

Substitution for $\cos u$ from $(1)$

$\ds $

$=$

$\ds -\paren {\sqrt {1 - x^2} - u x} + C$

Substitution for $\sin u$ from $(2)$

$\ds $

$=$

$\ds -\paren {\sqrt {1 - x^2} - x \arccos x} + C$

Substitution for $u$

$\ds $

$=$

$\ds x \arccos x - \sqrt {1 - x^2} + C$

simplifying

$\blacksquare$

Proof 2

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

$\ds u$

$=$

$\ds \arccos x$

$\ds \leadsto \ \ $

$\ds \frac {\d u} {\d x}$

$=$

$\ds \frac {-1} {\sqrt {a^2 - 1} }$

Derivative of $\arccos \dfrac x a$, setting $a := 1$

and let:

$\ds \frac {\d v} {\d x}$

$=$

$\ds 1$

$\ds \leadsto \ \ $

$\ds v$

$=$

$\ds x$

Primitive of Constant

Then:

$\ds \int \arccos x \rd x$

$=$

$\ds x \arccos x - \int x \paren {\frac {-1} {\sqrt {1 - x^2} } } \rd x + C$

Integration by Parts

$\ds $

$=$

$\ds x \arccos x + \int \frac {x \rd x} {\sqrt {1 - x^2} } + C$

simplifying

$\ds $

$=$

$\ds x \arccos x + \paren {-\sqrt {1 - x^2} } + C$

Primitive of $\dfrac x {\sqrt {1 - x^2} }$

$\ds $

$=$

$\ds x \arccos x - \sqrt {1 - x^2} + C$

simplifying

$\blacksquare$

Also see

Sources

1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals