Primitive of Minus One plus x Squared over One plus Fourth Power of x/Proof 2

Theorem

$\ds \int \frac {x^2 - 1} {x^4 + 1} \rd x = \frac 1 {2 \sqrt 2} \ln \size {\frac {x^2 - \sqrt 2 x + 1} {x^2 + \sqrt 2 x + 1} } + C$

Proof

\(\ds \int \frac {x^2 - 1} {x^4 + 1} \rd x\)

\(=\)

\(\ds \int \frac {x^2} {x^4 + 1} \rd x - \int \frac 1 {x^4 + 1} \rd x\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds \frac 1 {4 \sqrt 2} \map \ln {\frac {x^2 - x \sqrt 2 + 1} {x^2 + x \sqrt 2 + 1} } - \frac 1 {2 \sqrt 2} \paren {\map \arctan {1 - x \sqrt 2} - \map \arctan {1 + x \sqrt 2} }\)

Primitive of $\dfrac {x^2} {x^4 + a^4}$, setting $a = 1$

\(\ds \)

\(\)

\(\, \ds - \, \)

\(\ds \frac 1 {2 \sqrt 2} \paren {\map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + \frac 1 2 \ln \size {\frac {x^2 + \sqrt 2 x + 1} {x^2 - \sqrt 2 x + 1} } }\)

Primitive of $\dfrac 1 {1 + x^4}$

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