Primitives involving a squared minus x squared squared

Theorem

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

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

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

for $x^2 < a^2$.

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

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

for $x^2 < a^2$.

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

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

for $x^2 < a^2$.

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

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

for $x^2 < a^2$.

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

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

for $x^2 < a^2$.

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

$\ds \int \frac {\d x} {x^2 \paren {a^2 - x^2}^2} = \frac {-1} {a^4 x} + \frac x {2 a^4 \paren {a^2 - x^2} } + \frac 3 {4 a^5} \map \ln {\frac {a + x} {a - x} } + C$

for $x^2 < a^2$.

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

$\ds \int \frac {\d x} {x^3 \paren {a^2 - x^2}^2} = \frac {-1} {2 a^4 x^2} + \frac 1 {2 a^4 \paren {a^2 - x^2} } + \frac 1 {a^6} \map \ln {\frac {x^2} {a^2 - x^2} } + C$

for $x^2 < a^2$.

Also see

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