Primitive of Reciprocal of x fourth plus a fourth
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Theorem
- $\ds \int \frac {\d x} {x^4 + a^4} = \frac 1 {4 a^3 \sqrt 2} \map \ln {\frac {x^2 + a x \sqrt 2 + a^2} {x^2 - a x \sqrt 2 + a^2} } - \frac 1 {2 a^3 \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} }$
Proof
\(\ds \int \frac {\d x} {x^4 + a^4}\) | \(=\) | \(\ds \int \paren {\frac {x + a \sqrt 2} {2 a^3 \sqrt 2 \paren {x^2 + a x \sqrt 2 + a^2} } - \frac {x - a \sqrt 2} {2 a^3 \sqrt 2 \paren {x^2 - a x \sqrt 2 + a^2} } } \rd x\) | Partial Fraction Expansion | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \frac 1 {4 a^3 \sqrt 2} \int \frac {\paren {2 x + 2 a \sqrt 2} \rd x} {x^2 + a x \sqrt 2 + a^2} - \frac 1 {4 a^3 \sqrt 2} \int \frac {\paren {2 x - 2 a \sqrt 2} \rd x} {x^2 - a x \sqrt 2 + a^2}\) | Linear Combination of Primitives |
Then:
\(\ds \int \frac {\paren {2 x + 2 a \sqrt 2} \rd x} {x^2 + a x \sqrt 2 + a^2}\) | \(=\) | \(\ds \int \frac {\paren {2 x + a \sqrt 2} \rd x} {x^2 + a x \sqrt 2 + a^2} + a \sqrt 2 \int \frac {\d x} {x^2 + a x \sqrt 2 + a^2}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size {x^2 + a x \sqrt 2 + a^2} + a \sqrt 2 \int \frac {\d x} {x^2 + a x \sqrt 2 + a^2}\) | Primitive of Function under its Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size {x^2 + a x \sqrt 2 + a^2} + a \sqrt 2 \paren {\frac {\sqrt 2} a \, \map \arctan {1 + \frac {x \sqrt 2} a} }\) | Primitive of $\dfrac 1 {x^2 + a x \sqrt 2 + a^2}$ | |||||||||||
\(\text {(2)}: \quad\) | \(\ds \) | \(=\) | \(\ds \ln \size {x^2 + a x \sqrt 2 + a^2} + 2 \map \arctan {1 + \frac {x \sqrt 2} a}\) | simplifying |
Similarly:
\(\ds \int \frac {\paren {2 x - 2 a \sqrt 2} \rd x} {x^2 - a x \sqrt 2 + a^2}\) | \(=\) | \(\ds \int \frac {\paren {2 x - a \sqrt 2} \rd x} {x^2 - a x \sqrt 2 + a^2} - a \sqrt 2 \int \frac {\d x} {x^2 - a x \sqrt 2 + a^2}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size {x^2 - a x \sqrt 2 + a^2} - a \sqrt 2 \int \frac {\d x} {x^2 - a x \sqrt 2 + a^2}\) | Primitive of Function under its Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size {x^2 - a x \sqrt 2 + a^2} - a \sqrt 2 \paren {\frac {-\sqrt 2} a \, \map \arctan {1 - \frac {x \sqrt 2} a} }\) | Primitive of $\dfrac 1 {x^2 - a x \sqrt 2 + a^2}$ | |||||||||||
\(\text {(3)}: \quad\) | \(\ds \) | \(=\) | \(\ds \ln \size {x^2 - a x \sqrt 2 + a^2} + 2 \, \map \arctan {1 - \frac {x \sqrt 2} a}\) | simplifying |
Thus:
\(\ds \int \frac {\d x} {x^4 + a^4}\) | \(=\) | \(\ds \frac 1 {4 a^3 \sqrt 2} \int \frac {\paren {2 x + 2 a \sqrt 2} \rd x} {x^2 + a x \sqrt 2 + a^2} - \frac 1 {4 a^3 \sqrt 2} \int \frac {\paren {2 x - 2 a \sqrt 2} \rd x} {x^2 - a x \sqrt 2 + a^2}\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {4 a^3 \sqrt 2} \paren {\ln \size {x^2 + a x \sqrt 2 + a^2} + 2 \, \map \arctan {1 + \frac {x \sqrt 2} a} }\) | from $(2)$ | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac 1 {4 a^3 \sqrt 2} \paren {\ln \size {x^2 - a x \sqrt 2 + a^2} + 2 \, \map \arctan {1 - \frac {x \sqrt 2} a} }\) | from $(3)$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {4 a^3 \sqrt 2} \map \ln {\frac {x^2 + a x \sqrt 2 + a^2} {x^2 - a x \sqrt 2 + a^2} } - \frac 1 {2 a^3 \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} }\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^4 \pm a^4$: $14.311$