Primitives involving Root of x squared minus a squared

Theorem

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

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

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

for $x^2 > a^2$.

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

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

for $0 < a < \size x$.

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

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

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

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

for $x > a$.

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

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

Arccosine Form

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

for $0 < a < \size x$.

Arcsecant Form

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

for $0 < a < \size x$.

Arcsine Form

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

for $0 < a < \size x$.

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

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

for $\size x > a$.

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

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

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

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

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

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

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

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

for $\size x \ge a$.

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

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

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

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

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

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

for $x^2 \ge a^2$.

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

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

Also see

• Primitives involving $\left({\sqrt{x^2 - a^2} }\right)^3$

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