Primitive of x cubed over x fourth plus a fourth
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Theorem
- $\ds \int \frac {x^3 \rd x} {x^4 + a^4} = \frac {\map \ln {x^4 + a^4} } 4 + C$
Proof 1
\(\ds \frac \d {\d x} x^4\) | \(=\) | \(\ds 4 x^3\) | Primitive of Power | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {x^3 \rd x} {x^4 + a^4}\) | \(=\) | \(\ds \frac 1 4 \ln \size {x^4 + a^4} + C\) | Primitive of Function under its Derivative | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \ln {x^4 + a^4} } 4 + C\) | Absolute Value of Even Power‎ |
$\blacksquare$
Proof 2
From Primitive of $\dfrac {x^{n - 1} } {x^n + a^n}$:
- $\ds \int \frac {x^{n - 1} \rd x} {x^n + a^n} = \frac 1 n \ln \size {x^n + a^n} + C$
So:
\(\ds \int \frac {x^3 \rd x} {x^4 + a^4}\) | \(=\) | \(\ds \frac 1 4 \ln \size {x^4 + a^4} + C\) | Primitive of $\dfrac {x^{n - 1} } {\paren {x^n + a^n} }$ with $n = 4$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \ln {x^4 + a^4} } 4 + C\) | Absolute Value of Even Power |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^4 \pm a^4$: $14.314$