Primitive of x cubed over Root of x squared minus a squared cubed
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Theorem
- $\ds \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \sqrt {x^2 - a^2} - \frac {a^2} {\sqrt {x^2 - a^2} } + C$
for $\size x > a$.
Proof
\(\ds \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 - a^2} }^3}\) | \(=\) | \(\ds \int \frac {x \paren {x^2} \rd x} {\paren {\sqrt {x^2 - a^2} }^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {x \paren {x^2 - a^2 + a^2} \rd x} {\paren {\sqrt {x^2 - a^2} }^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {x \paren {x^2 - a^2} \rd x} {\paren {\sqrt {x^2 - a^2} }^3} + a^2 \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {x \rd x} {\sqrt {x^2 - a^2} } + a^2 \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {x^2 - a^2} + a^2 \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3} + C\) | Primitive of $\dfrac x {\sqrt {x^2 - a^2} }$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {x^2 - a^2} + a^2 \frac {-1} {\sqrt {x^2 - a^2} } + C\) | Primitive of $\dfrac x {\paren {\sqrt {x^2 - a^2} }^3}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {x^2 - a^2} - \frac {a^2} {\sqrt {x^2 - a^2} } + C\) | simplifying |
$\blacksquare$
Also see
- Primitive of $\dfrac {x^3} {\paren {\sqrt {x^2 + a^2} }^3}$
- Primitive of $\dfrac {x^3} {\paren {\sqrt {a^2 - x^2} }^3}$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.226$