Primitive of Reciprocal of x squared plus a squared squared

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Theorem

$\ds \int \frac {\d x} {\paren {x^2 + a^2}^2} = \frac x {2 a^2 \paren {x^2 + a^2} } + \frac 1 {2 a^3} \arctan \frac x a + C$


Proof

Let:

\(\ds x\) \(=\) \(\ds a \tan \theta\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d x} {\d \theta}\) \(=\) \(\ds a \sec^2 \theta\) Derivative of Tangent Function
\(\ds \leadsto \ \ \) \(\ds \int \frac {\d x} {\paren {x^2 + a^2}^2}\) \(=\) \(\ds \int \frac {a \sec^2 \theta} {\paren {a^2 \tan^2 \theta + a^2}^2} \rd \theta\) Integration by Substitution
\(\ds \) \(=\) \(\ds \int \frac {a \sec^2 \theta} {\paren {a^2 \sec^2 \theta}^2} \rd \theta\) Difference of Squares of Secant and Tangent
\(\ds \) \(=\) \(\ds \frac 1 {a^3} \int \frac {\d \theta} {\sec^2 \theta}\) simplification
\(\ds \) \(=\) \(\ds \frac 1 {a^3} \int \cos^2 \theta \rd \theta\) Definition of Real Secant Function
\(\ds \) \(=\) \(\ds \frac 1 {a^3} \paren {\frac \theta 2 + \frac {\sin 2 \theta} 4} + C\) Primitive of Square of Cosine Function
\(\ds \) \(=\) \(\ds \frac \theta {2 a^3} + \frac 1 {2 a^3} \frac {\tan \theta} {1 + \tan^2 \theta} + C\) Tangent Half-Angle Substitution for Sine
\(\ds \) \(=\) \(\ds \frac x {2 a^2 \paren {x^2 + a^2} } + \frac 1 {2 a^3} \arctan \frac x a + C\) substituting for $\theta$

$\blacksquare$


Also see


Sources

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