Primitives involving a squared minus x squared

Theorem

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

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

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

where $\size x < a$.

Primitive of Reciprocal of $a^2 - x^2$: Logarithm Form

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

Primitive $x$ over $a^2 - x^2$

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

for $x^2 < a^2$.

Theorem

Logarithm Form

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

for $x^2 < a^2$.

Inverse Hyperbolic Tangent Form

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

for $x^2 < a^2$.

Also see

• Primitive of $\dfrac {x^2} {x^2 - a^2}$

Primitive $x^3$ over $a^2 - x^2$

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

for $x^2 < a^2$.

Primitive of Reciprocal of $x \left({a^2 - x^2}\right)$

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

for $x^2 < a^2$.

Primitive of Reciprocal of $x^2 \left({a^2 - x^2}\right)$

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

for $x^2 < a^2$.

Primitive of Reciprocal of $x^3 \left({a^2 - x^2}\right)$

$\ds \int \frac {\d x} {x^3 \paren {a^2 - x^2} } = \frac {-1} {2 a^2 x^2} + \frac 1 {2 a^4} \map \ln {\frac {x^2} {a^2 - x^2} } + C$

for $x^2 < a^2$.

Also see

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