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

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Proof

\(\ds \int \frac {x^3 \rd x} {\paren {\sqrt {a^2 - x^2} }^3}\) \(=\) \(\ds \int \frac {x \paren {x^2} \rd x} {\paren {\sqrt {a^2 - x^2} }^3}\)
\(\ds \) \(=\) \(\ds \int \frac {x \paren {x^2 - a^2 + a^2} \rd x} {\paren {\sqrt {a^2 - x^2} }^3}\)
\(\ds \) \(=\) \(\ds -\int \frac {x \paren {a^2 - x^2} \rd x} {\paren {\sqrt {a^2 - x^2} }^3} + a^2 \int \frac {x \rd x} {\paren {\sqrt {a^2 - x^2} }^3}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds -\int \frac {x \rd x} {\sqrt {a^2 - x^2} } + a^2 \int \frac {x \rd x} {\paren {\sqrt {a^2 - x^2} }^3}\) simplifying
\(\ds \) \(=\) \(\ds \sqrt {a^2 - x^2} + a^2 \int \frac {x \rd x} {\paren {\sqrt {a^2 - x^2} }^3} + C\) Primitive of $\dfrac x {\sqrt {a^2 - x^2} }$
\(\ds \) \(=\) \(\ds \sqrt {a^2 - x^2} + a^2 \frac 1 {\sqrt {a^2 - x^2} } + C\) Primitive of $\dfrac x {\paren {\sqrt {a^2 - x^2} }^3}$
\(\ds \) \(=\) \(\ds \sqrt {a^2 - x^2} + \frac {a^2} {\sqrt {a^2 - x^2} } + C\) simplifying

$\blacksquare$


Also see


Sources