Primitive of Power of x by Arccosine of x over a/Proof 2
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Theorem
- $\ds \int x^m \arccos \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arccos \frac x a + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {a^2 - x^2} }$
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:
| \(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds x^m\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {x^{m + 1} } {m + 1}\) | Primitive of Power |
Then:
| \(\ds \int x^m \arccos \frac x a \rd x\) | \(=\) | \(\ds \frac {x^{m + 1} } {m + 1} \arccos \frac x a - \int \frac {x^{m + 1} } {m + 1} \paren {\frac {-1} {\sqrt {a^2 - x^2} } } \rd x\) | Integration by Parts | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {x^{m + 1} } {m + 1} \arccos \frac x a + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {a^2 - x^2} }\) | Primitive of Constant Multiple of Function |
$\blacksquare$