Primitive of Function of Arccosine
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Theorem
- $\ds \int \map F {\arccos \frac x a} \rd x = -a \int \map F u \sin u \rd u$
where $u = \arccos \dfrac x a$.
Proof
First note that:
| \(\ds u\) | \(=\) | \(\ds \arccos \frac x a\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds a \cos u\) | Definition of Real Arccosine |
Then:
| \(\ds u\) | \(=\) | \(\ds \arccos \frac x a\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \frac {-1} {\sqrt {a^2 - x^2} }\) | Derivative of Arccosine Function: Corollary | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \int \map F {\arccos \frac x a} \rd x\) | \(=\) | \(\ds \int \map F u \paren {-\sqrt {a^2 - x^2} } \rd u\) | Primitive of Composite Function | ||||||||||
| \(\ds \) | \(=\) | \(\ds \int \map F u \paren {-\sqrt {a^2 - a^2 \cos^2 u} } \rd u\) | Definition of $x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \map F u a \paren {-\sqrt {1 - \cos^2 u} } \rd u\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \map F u a \paren {-\sin u} \rd u\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds -a \int\map F u \sin u \rd u\) | Primitive of Constant Multiple of Function |
$\blacksquare$