Primitive of Reciprocal of x by Root of a squared minus x squared cubed
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Theorem
- $\ds \int \frac {\d x} {x \paren {\sqrt {a^2 - x^2} }^3} = \frac 1 {a^2 \sqrt {a^2 - x^2} } - \frac 1 {a^3} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$
Proof
Let:
\(\ds z\) | \(=\) | \(\ds x^2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds 2 x\) | Power Rule for Derivatives | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {x \paren {\sqrt {a^2 - x^2} }^3}\) | \(=\) | \(\ds \int \frac {\d z} {2 \sqrt z \sqrt z \paren {\sqrt {a^2 - z} }^3}\) | Integration by Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int \frac {\d z} {z \paren {\sqrt {a^2 - z} }^3}\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\frac 2 {a^2 \sqrt {a^2 - z} } + \frac 1 {a^2} \int \frac {\d z} {z \sqrt {a^2 - z} } } + C\) | Primitive of $\dfrac 1 {x \paren {\sqrt {a x + b} }^m }$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2 \sqrt {a^2 - z} } + \frac 1 {2 a^2} \int \frac {\d z} {z \sqrt {a^2 - z} } + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2 \sqrt {a^2 - x^2} } + \frac 1 {2 a^2} \int \frac {2 x \rd x} {x^2 \sqrt {a^2 - x^2} } + C\) | substituting for $z$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2 \sqrt {a^2 - x^2} } + \frac 1 {a^2} \int \frac {\d x} {x \sqrt {a^2 - x^2} } + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac1 {a^2 \sqrt {a^2 - x^2} } - \frac 1 {a^3} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C\) | Primitive of $\dfrac 1 {x \sqrt {a^2 - x^2} }$ |
$\blacksquare$
Also see
- Primitive of $\dfrac 1 {x \paren {\sqrt {x^2 + a^2} }^3}$
- Primitive of $\dfrac 1 {x \paren {\sqrt {x^2 - a^2} }^3}$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {a^2 - x^2}$: $14.255$