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

$\blacksquare$


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