Primitive of x over Root of a squared minus x squared cubed

Theorem

$\ds \int \frac {x \rd x} {\paren {\sqrt {a^2 - x^2} }^3} = \frac 1 {\sqrt {a^2 - x^2} } + C$

Proof

Let:

\(\ds z\)

\(=\)

\(\ds a^2 - x^2\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\d z} {\d x}\)

\(=\)

\(\ds -2 x\)

Power Rule for Derivatives

\(\ds \leadsto \ \ \)

\(\ds \int \frac {x \rd x} {\paren {\sqrt {a^2 - x^2} }^3}\)

\(=\)

\(\ds \int \frac {x \rd z} {-2 x z^{3/2} }\)

Integration by Substitution

\(\ds \)

\(=\)

\(\ds \frac 1 2 \int -z^{-3/2} \rd z\)

Primitive of Constant Multiple of Function

\(\ds \)

\(=\)

\(\ds \frac 1 2 \paren {\frac {z^{1/2} } {\frac 1 2} } + C\)

Primitive of Power

\(\ds \)

\(=\)

\(\ds \frac 1 {\sqrt z} + C\)

simplifying

\(\ds \)

\(=\)

\(\ds \frac 1 {\sqrt {a^2 - x^2} } + C\)

substituting for $z$

$\blacksquare$

Also see

• Primitive of $\dfrac x {\paren {\sqrt {x^2 + a^2} }^3}$
• Primitive of $\dfrac x {\paren {\sqrt {x^2 - a^2} }^3}$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {a^2 - x^2}$: $14.252$