Primitive of Reciprocal of x fourth minus a fourth
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Theorem
- $\ds \int \frac {\d x} {x^4 - a^4} = \frac 1 {4 a^3} \ln \size {\frac {x - a} {x + a} } - \frac 1 {2 a^3} \arctan \frac x a$
Proof
\(\ds \int \frac {\d x} {x^4 - a^4}\) | \(=\) | \(\ds \int \frac {\d x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }\) | Difference of Two Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {a^2 \rd x} {a^2 \paren {x^2 + a^2} \paren {x^2 - a^2} }\) | multiplying top and bottom by $a^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 \paren {x^2 + a^2} \paren {x^2 - a^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \int \frac {\paren {x^2 + a^2} \d x} {\paren {x^2 + a^2} \paren {x^2 - a^2} } - \frac 1 {a^2} \int \frac {x^2 \rd x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {x^2 \rd x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\paren {x^2 - a^2 + a^2} \rd x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }\) | Difference of Two Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\paren {x^2 - a^2} \rd x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac 1 {a^2} \int \frac {a^2 \rd x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\d x} {x^2 + a^2} - \int \frac {\d x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\d x} {x^2 + a^2} - \int \frac {\d x} {x^4 - a^4}\) | Difference of Two Squares | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \int \frac {\d x} {x^4 - a^4}\) | \(=\) | \(\ds \frac 1 {a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\d x} {x^2 + a^2}\) | gathering terms | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {x^4 - a^4}\) | \(=\) | \(\ds \frac 1 {2 a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {2 a^2} \int \frac {\d x} {x^2 + a^2}\) | gathering terms | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {4 a^3} \ln \size {\frac {x - a} {x + a} } - \frac 1 {2 a^2} \int \frac {\d x} {x^2 + a^2}\) | Primitive of $\dfrac 1 {x^2 - a^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {4 a^3} \ln \size {\frac {x - a} {x + a} } - \frac 1 {2 a^3} \arctan \frac x a\) | Primitive of $\dfrac 1 {x^2 + a^2}$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^4 \pm a^4$: $14.318$