Primitive of Function of Arccosine

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Theorem

$\displaystyle \int F \paren {\arccos \frac x a} \rd x = -a \int F \paren u \sin u \rd u$

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


Proof

First note that:

\(\displaystyle u\) \(=\) \(\displaystyle \arccos \frac x a\)
\(\displaystyle \implies \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle a \cos u\) Definition of Arccosine


Then:

\(\displaystyle u\) \(=\) \(\displaystyle \arccos \frac x a\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\d u} {\d x}\) \(=\) \(\displaystyle \frac {-1} {\sqrt {a^2 - x^2} }\) Derivative of Arccosine Function: Corollary
\(\displaystyle \implies \ \ \) \(\displaystyle \int F \paren {\arccos \frac x a} \rd x\) \(=\) \(\displaystyle \int F \paren u \ \paren {-\sqrt {a^2 - x^2} } \rd u\) Primitive of Composite Function
\(\displaystyle \) \(=\) \(\displaystyle \int F \paren u \ \paren {-\sqrt {a^2 - a^2 \cos^2 u} } \rd u\) Definition of $x$
\(\displaystyle \) \(=\) \(\displaystyle \int F \paren u \ a \paren {-\sqrt {1 - \cos^2 u} } \rd u\)
\(\displaystyle \) \(=\) \(\displaystyle \int F \paren u \ a \paren {-\sin u} \rd u\) Sum of Squares of Sine and Cosine
\(\displaystyle \) \(=\) \(\displaystyle -a \int F \paren u \ \sin u \rd u\) Primitive of Constant Multiple of Function

$\blacksquare$


Also see