Primitive of Function of Arctangent

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Theorem

$\ds \int \map F {\arctan \frac x a} \rd x = a \int \map F u \sec^2 u \rd u$

where $u = \arctan \dfrac x a$.


Proof

First note that:

\(\ds u\) \(=\) \(\ds \arctan \frac x a\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds a \tan u\) Definition of Arctangent


Then:

\(\ds u\) \(=\) \(\ds \arctan \frac x a\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac a {a^2 + x^2}\) Derivative of Arctangent Function: Corollary
\(\ds \leadsto \ \ \) \(\ds \int \map F {\arctan \frac x a} \rd x\) \(=\) \(\ds \int \map F u \frac {a^2 + x^2} a \rd u\) Primitive of Composite Function
\(\ds \) \(=\) \(\ds \int \map F u \frac {a^2 + a^2 \tan^2 u} a \rd u\) Definition of $x$
\(\ds \) \(=\) \(\ds \int \map F u a^2 \frac {1 + \tan^2 u} a \rd u\)
\(\ds \) \(=\) \(\ds \int \map F u a \sec^2 u \rd u\) Difference of Squares of Secant and Tangent
\(\ds \) \(=\) \(\ds a \int \map F u \sec^2 u \rd u\) Primitive of Constant Multiple of Function

$\blacksquare$


Also see