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

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Theorem

$\displaystyle \int \frac {x^2 \ \mathrm d x} {\sqrt {a^2 - x^2} } = \frac {-x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a + C$


Proof

With a view to expressing the problem in the form:

$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \mathrm d x$

let:

\(\displaystyle u\) \(=\) \(\displaystyle x\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d u}{\mathrm d x}\) \(=\) \(\displaystyle 1\) Power Rule for Derivatives


and let:

\(\displaystyle \frac {\mathrm d v}{\mathrm d x}\) \(=\) \(\displaystyle \frac x {\sqrt {a^2 - x^2} }\)
\(\displaystyle \implies \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle -\sqrt {a^2 - x^2}\) Primitive of $\dfrac x {\sqrt {a^2 - x^2} }$


Then:

\(\displaystyle \int \frac {x^2 \ \mathrm d x} {\sqrt {a^2 - x^2} }\) \(=\) \(\displaystyle \int x \frac {x \ \mathrm d x} {\sqrt {a^2 - x^2} }\)
\(\displaystyle \) \(=\) \(\displaystyle -x \sqrt {a^2 - x^2} - \int \left({-\sqrt {a^2 - x^2} }\right) \ \mathrm d x\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle -x \sqrt {a^2 - x^2} + \int \left({\sqrt {a^2 - x^2} }\right) \ \mathrm d x\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle -x \sqrt {a^2 - x^2} + \left({\frac {x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a}\right) + C\) Primitive of $\sqrt {a^2 - x^2}$
\(\displaystyle \) \(=\) \(\displaystyle \frac {-x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a + C\)

$\blacksquare$


Also see


Sources