Primitive of x by Root of a squared minus x squared

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Proof

Let:

\(\ds z\) \(=\) \(\ds x^2\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d z} {\d x}\) \(=\) \(\ds 2 x\) Power Rule for Derivatives
\(\ds \leadsto \ \ \) \(\ds \int x \sqrt {a^2 - x^2} \rd x\) \(=\) \(\ds \int \frac {\sqrt z \sqrt {a^2 - z} \rd z} {2 \sqrt z}\) Integration by Substitution
\(\ds \) \(=\) \(\ds \frac 1 2 \int \sqrt {a^2 - z} \rd z\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds \frac 1 2 \frac {2 \paren {\sqrt {a^2 - z} }^3} {-3} + C\) Primitive of $\sqrt {a x + b}$
\(\ds \) \(=\) \(\ds \frac {-\paren {\sqrt {a^2 - x^2} }^3} 3 + C\) substituting for $z$ and simplifying

$\blacksquare$


Also see


Sources