Primitive of Arccosine of x over a/Proof 2
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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
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 |
$\blacksquare$