Primitive of Arccosine of x over a

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Theorem

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


Proof 1

Let:

\(\ds u\) \(=\) \(\ds \arccos \frac x a\)
\(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \cos u\) \(=\) \(\ds \frac x a\) Definition of Real Arccosine
\(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \cos u\) \(=\) \(\ds \sqrt {1 - \frac {x^2} {a^2} }\) Sum of Squares of Sine and Cosine


Then:

\(\ds \int \arccos \frac x a \rd x\) \(=\) \(\ds -a \int u \sin u \rd u\) Primitive of Function of Arccosine
\(\ds \) \(=\) \(\ds -a \paren {\sin u - u \cos u} + C\) Primitive of $x \sin a x$
\(\ds \) \(=\) \(\ds -a \paren {\sin u - u \frac x a} + C\) Substitution for $\cos u$ from $(1)$
\(\ds \) \(=\) \(\ds -a \paren {\sqrt {1 - \frac {x^2} {a^2} } - u \frac x a} + C\) Substitution for $\sin u$ from $(2)$
\(\ds \) \(=\) \(\ds -a \paren {\sqrt {1 - \frac {x^2} {a^2} } - \frac x a \arccos \frac x a} + C\) Substitution for $u$
\(\ds \) \(=\) \(\ds x \arccos \frac x a - \sqrt {a^2 - x^2} + C\) 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:

\(\ds u\) \(=\) \(\ds \arccos \frac x a\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac {-1} {\sqrt {a^2 - x^2} }\) Derivative of $\arccos \dfrac x a$


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds 1\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds x\) Primitive of Constant


Then:

\(\ds \int \arccos \frac x a \rd x\) \(=\) \(\ds x \arccos \frac x a - \int x \paren {\frac {-1} {\sqrt {a^2 - x^2} } } \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds x \arccos \frac x a + \int \frac {x \rd x} {\sqrt {a^2 - x^2} } + C\) simplifying
\(\ds \) \(=\) \(\ds x \arccos \frac x a + \paren {-\sqrt {a^2 - x^2} } + C\) Primitive of $\dfrac x {\sqrt {a^2 - x^2} }$
\(\ds \) \(=\) \(\ds x \arccos \frac x a - \sqrt {a^2 - x^2} + C\) simplifying

$\blacksquare$


Proof 3

\(\ds \int \arccos x \rd x\) \(=\) \(\ds x \arccos x - \sqrt {1 - x^2} + C\) Primitive of $\arccos x$
\(\ds \leadsto \ \ \) \(\ds \int \arccos \frac x a \rd x\) \(=\) \(\ds \frac 1 {1 / a} \paren {\frac x a \arccos \frac x a - \sqrt {1 - \paren {\dfrac x a}^2} } + C\) Primitive of Function of Constant Multiple
\(\ds \) \(=\) \(\ds x \arccos \frac x a - \sqrt {a^2 - x^2} + C\) after simplifying

$\blacksquare$


Also see


Sources

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