Primitives involving Root of a squared minus x squared

Theorem

This page gathers together the primitives of some expressions involving $\sqrt {a^2 - x^2}$.

Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$: $\arcsin$ form

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

Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$: $\arccos$ form

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

Primitive of $x$ over $\sqrt {a^2 - x^2}$

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

Primitive of $x^2$ over $\sqrt {a^2 - x^2}$

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

\(=\)

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

\(\ds \)

\(=\)

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

either way round, whichever you prefer

Primitive of $x^3$ over $\sqrt {a^2 - x^2}$

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

Primitive of Reciprocal of $x \sqrt {a^2 - x^2}$: $\sech^{-1}$ form

For $a > 0$ and $0 < \size x < a$:

$\ds \int \frac {\d x} {x \sqrt {a^2 - x^2} } = -\frac 1 a \sech^{-1} {\frac {\size x} a} + C$

Primitive of Reciprocal of $x \sqrt {a^2 - x^2}$: Logarithm Form

For $a > 0$ and $0 < \size x < a$:

$\ds \int \frac {\d x} {x \sqrt {a^2 - x^2} } = -\frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$

Primitive of Reciprocal of $x^2 \sqrt {a^2 - x^2}$

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

Primitive of Reciprocal of $x^3 \sqrt {a^2 - x^2}$

$\ds \int \frac {\d x} {x^3 \sqrt {a^2 - x^2} } = \frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} - \frac 1 {2 a^3} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$

Theorem

Arcsine Form

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

Arccosine Form

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

Primitive of $x \sqrt {a^2 - x^2}$

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

Primitive of $x^2 \sqrt {a^2 - x^2}$

$\ds \int x^2 \sqrt {a^2 - x^2} \rd x = -\frac {x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {a^2 x \sqrt {a^2 - x^2} } 8 + \frac {a^4} 8 \arcsin \frac x a + C$

Primitive of $x^3 \sqrt {a^2 - x^2}$

$\ds \int x^3 \sqrt {a^2 - x^2} \rd x = \frac {\paren {\sqrt {a^2 - x^2} }^5} 5 - \frac {a^2 \paren {\sqrt {a^2 - x^2} }^3} 3 + C$

Primitive of $\sqrt {a^2 - x^2}$ over $x$

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

Primitive of $\sqrt {a^2 - x^2}$ over $x^2$

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

Primitive of $\sqrt {a^2 - x^2}$ over $x^3$

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

Also see

• Primitives involving $\paren {\sqrt{a^2 - x^2} }^3$

• Primitives involving $\sqrt {x^2 + a^2}$
• Primitives involving $\sqrt {x^2 - a^2}$