Primitive of x squared over x squared plus a squared squared

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Theorem

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


Proof 1

Let:

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

$\blacksquare$


Proof 2

\(\ds \int \frac {x^2 \rd x} {\paren {x^2 + a^2}^2}\) \(=\) \(\ds \int \frac {x^2 + a^2 - a^2} {\paren {x^2 + a^2}^2} \rd x\)
\(\ds \) \(=\) \(\ds \int \frac {x^2 + a^2} {\paren {x^2 + a^2}^2} \rd x - a^2 \int \frac {\d x} {\paren {x^2 + a^2}^2}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \int \frac {\d x} {x^2 + a^2} + a^2 \int \frac {\d x} {\paren {x^2 + a^2}^2}\) simplification
\(\ds \) \(=\) \(\ds \frac 1 a \arctan {\frac x a} + a^2 \int \frac {\d x} {\paren {x^2 + a^2}^2} + C\) Primitive of Reciprocal of $x^2 + a^2$
\(\ds \) \(=\) \(\ds \frac 1 a \arctan {\frac x a} + a^2 \paren {\frac x {2 a^2 \paren {x^2 + a^2} } + \frac 1 {2 a^3} \arctan \frac x a} + C\) Primitive of Reciprocal of $\paren {x^2 + a^2}^2$
\(\ds \) \(=\) \(\ds \frac {-x} {2 \paren {x^2 + a^2} } + \frac 1 {2 a} \arctan \frac x a + C\) simplifying

$\blacksquare$


Sources

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