Primitive of Root of 2 a x minus x squared over x squared

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int \dfrac {\sqrt {2 a x - x^2} } {x^2} \rd x = -2 \sqrt {\dfrac {2 a - x} x} - \arcsin \dfrac {x - a} a + C$


Proof

With a view to expressing the primitive 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:

\(\ds u\) \(=\) \(\ds \sqrt {2 a x - x^2}\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \dfrac {2 a - 2 x} {2 \sqrt {2 a x - x^2} }\) Derivative of Power, Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \dfrac {a - x} {\sqrt {2 a x - x^2} }\) simplifying


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \dfrac 1 {x^2}\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds -\dfrac 1 x\) Derivative of Power


Then:

\(\ds \int \dfrac {\sqrt {2 a x - x^2} } {x^2} \rd x\) \(=\) \(\ds \int u \frac {\d v} {\d x} \rd x\)
\(\ds \) \(=\) \(\ds \paren {\sqrt {2 a x - x^2} } \paren {-\dfrac 1 x} - \int \paren {-\dfrac 1 x} \paren {\dfrac {a - x} {\sqrt {2 a x - x^2} } } \rd x\) Integration by Parts
\(\ds \) \(=\) \(\ds -\sqrt {\dfrac {x \paren {2 a - x} } {x^2} } + \int \dfrac {a - x} {x \sqrt {2 a x - x^2} } \rd x\) simplifying
\(\ds \) \(=\) \(\ds -\sqrt {\dfrac {2 a - x} x} + a \int \dfrac 1 {x \sqrt {2 a x - x^2} } \rd x - \int \dfrac 1 {\sqrt {2 a x - x^2} } \rd x\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds -\sqrt {\dfrac {2 a - x} x} + a \paren {-\dfrac 1 a \sqrt {\dfrac {2 a - x} x} } - \int \dfrac 1 {\sqrt {2 a x - x^2} } \rd x\) Primitive of $\dfrac 1 {x \sqrt {2 a x - x^2} }$
\(\ds \) \(=\) \(\ds -\sqrt {\dfrac {2 a - x} x} - \sqrt {\dfrac {2 a - x} x} - \arcsin \dfrac {x - a} a + C\) Primitive of $\dfrac 1 {\sqrt {2 a x - x^2} }$
\(\ds \) \(=\) \(\ds -2 \sqrt {\dfrac {2 a - x} x} - \arcsin \dfrac {x - a} a + C\) simplifying

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Primitive_of_Root_of_2_a_x_minus_x_squared_over_x_squared&oldid=675152"