Primitive of x squared over Root of x squared plus a squared cubed
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Theorem
- $\ds \int \frac {x^2 \rd x} {\paren {\sqrt {x^2 + a^2} }^3} = \frac {-x} {\sqrt {x^2 + a^2} } + \map \ln {x + \sqrt {x^2 + a^2} } + C$
Proof
With a view to expressing the problem 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:
\(\ds u\) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds 1\) | Power Rule for Derivatives |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \frac x {\paren {\sqrt {x^2 + a^2} }^3}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {-1} {\sqrt {x^2 + a^2} }\) | Primitive of $\dfrac x {\paren {\sqrt {x^2 + a^2} }^3}$ |
Then:
\(\ds \int \frac {x^2 \rd x} {\paren {\sqrt {x^2 + a^2} }^3}\) | \(=\) | \(\ds \int x \frac {x \rd x} {\paren {\sqrt {x^2 + a^2} }^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x \frac {-1} {\sqrt {x^2 + a^2} } - \int \frac {-1} {\sqrt {x^2 + a^2} } 1 + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-x} {\sqrt {x^2 + a^2} } + \int \frac 1 {\sqrt {x^2 + a^2} } + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-x} {\sqrt {x^2 + a^2} } + \map \ln {x + \sqrt {x^2 + a^2} } + C\) | Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$ |
$\blacksquare$
Also see
- Primitive of $\dfrac {x^2} {\paren {\sqrt {x^2 - a^2} }^3}$
- Primitive of $\dfrac {x^2} {\paren {\sqrt {a^2 - x^2} }^3}$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.198$