Primitive of Reciprocal of x squared by a squared minus x squared squared
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Theorem
- $\ds \int \frac {\d x} {x^2 \paren {a^2 - x^2}^2} = \frac {-1} {a^4 x} + \frac x {2 a^4 \paren {a^2 - x^2} } + \frac 3 {4 a^5} \map \ln {\frac {a + x} {a - x} } + C$
for $x^2 < a^2$.
Proof
| \(\ds \int \frac {\d x} {x^2 \paren {a^2 - x^2}^2}\) | \(=\) | \(\ds \int \paren {\frac 1 {a^4 x^2} + \frac 3 {4 a^5 \paren {a + x} } + \frac 3 {4 a^5 \paren {a - x} } + \frac 1 {4 a^4 \paren {a + x}^2} + \frac 1 {4 a^4 \paren {a - x}^2} } \rd x\) | Partial Fraction Expansion | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {a^4} \int \frac {\d x} {x^2} + \frac 3 {4 a^5} \int \frac {\d x} {a + x} + \frac 3 {4 a^5} \int \frac {\d x} {a - x} + \frac 1 {4 a^4} \int \frac {\d x} {\paren {a + x}^2} + \frac 1 {4 a^4} \int \frac {\d x} {\paren {a - x}^2}\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {a^4} \frac {-1} x + \frac 3 {4 a^5} \int \frac {\d x} {a + x} + \frac 3 {4 a^5} \int \frac {\d x} {a - x} + \frac 1 {4 a^4} \int \frac {\d x} {\paren {a + x}^2} + \frac 1 {4 a^4} \int \frac {\d x} {\paren {a - x}^2} + C\) | Primitive of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-1} {a^4 x} + \frac 3 {4 a^5} \ln \size {a + x} + \frac 3 {4 a^5} \paren {-\ln \size {a - x} } + \frac 1 {4 a^4} \int \frac {\d x} {\paren {a + x}^2} + \frac 1 {4 a^4} \int \frac {\d x} {\paren {a - x}^2} + C\) | Primitive of Reciprocal | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-1} {a^4 x} + \frac 3 {4 a^5} \ln \size {a + x} - \frac 3 {4 a^5} \ln \size {a - x} + \frac 1 {4 a^4} \frac {-1} {a + x} + \frac 1 {4 a^4} \frac 1 {a - x} + C\) | Primitive of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-1} {a^4 x} + \frac 3 {4 a^5} \ln \size {\frac {a + x} {a - x} } + \frac {-1} {4 a^4 \paren {a + x} } + \frac 1 {4 a^4 \paren {a - x} } + C\) | Difference of Logarithms | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-1} {a^4 x} + \frac 3 {4 a^5} \map \ln {\frac {a + x} {a - x} } + \frac {-1} {4 a^4 \paren {a + x} } + \frac 1 {4 a^4 \paren {a - x} } + C\) | Sign of Quotient of Factors of Difference of Squares | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-1} {a^4 x} + \frac x {2 a^4 \paren {a^2 - x^2} } + \frac 3 {4 a^5} \map \ln {\frac {a + x} {a - x} } + C\) | simplification |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.175$