Primitive of x squared over x squared plus a squared squared/Proof 1

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

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 $\theta$

$\blacksquare$