Primitives involving Root of x squared plus a squared

Theorem

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

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

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

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

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

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

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

Inverse Hyperbolic Sine Form

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

Logarithm Form

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

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

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

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

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

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

For $x \in \R_{\ne 0}$:

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

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

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

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

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

Inverse Hyperbolic Sine Form

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

Logarithm Form

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

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 $x^2 \sqrt {x^2 + a^2}$

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

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

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

Logarithm Form

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

Inverse Hyperbolic Sine Form

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

Logarithm Form

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

Inverse Hyperbolic Sine Form

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

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

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

Also see

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

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