Primitive of Reciprocal of Power of Root of 2 a x minus x squared
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Theorem
- $\ds \int \dfrac {\d x} {\paren {\sqrt {2 a x - x^2} }^n} = \frac {\paren {x - a} \paren {\sqrt {2 a x - x^2} }^{2 - n} } {\paren {n - 2} a^2} + \frac {n - 3} {\paren {n - 2} a^2} \int \dfrac {\d x} {\paren {\sqrt {2 a x - x^2} }^{n - 2} }$
Proof
| \(\ds \int \dfrac {\d x} {\paren {\sqrt {2 a x - x^2} }^{n - 2} }\) | \(=\) | \(\ds \int \paren {\sqrt {2 a x - x^2} }^{2 - n} \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\paren {x - a} \paren {\sqrt {2 a x - x^2} }^{2 - n} } {\paren {2 - n} + 1} + \frac {\paren {2 - n} a^2} {\paren {2 - n} + 1} \int \paren {\sqrt {2 a x - x^2} }^{\paren {2 - n} - 2} \rd x\) | Primitive of $\paren {\sqrt {2 a x - x^2} }^{n - 2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\paren {x - a} \paren {\sqrt {2 a x - x^2} }^{2 - n} } {3 - n} + \frac {\paren {2 - n} a^2} {3 - n} \int \dfrac {\d x} {\paren {\sqrt {2 a x - x^2} }^n} \rd x\) | simplifying | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {n - 3} {\paren {n - 2} a^2} \int \dfrac {\d x} {\paren {\sqrt {2 a x - x^2} }^{n - 2} }\) | \(=\) | \(\ds \frac {n - 3} {\paren {n - 2} a^2} \frac {\paren {x - a} \paren {\sqrt {2 a x - x^2} }^{2 - n} } {3 - n} + \int \dfrac {\d x} {\paren {\sqrt {2 a x - x^2} }^n} \rd x\) | multiplying both sides by $\dfrac {n - 3} {\paren {n - 2} a^2}$ | ||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac {\paren {x - a} \paren {\sqrt {2 a x - x^2} }^{2 - n} } {\paren {n - 2} a^2} + \int \dfrac {\d x} {\paren {\sqrt {2 a x - x^2} }^n} \rd x\) | simplifying |
from which the result follows immediately.
$\blacksquare$
Sources
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $50$.