Primitive of Logarithm of x squared plus a squared

Theorem

 * $\displaystyle \int \map \ln {x^2 + a^2} \rd x = x \map \ln {x^2 + a^2} - 2 x + 2 a \arctan \frac x a + C$

Proof
With a view to expressing the primitive in the form:
 * $\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

and let:

Then:

Also see

 * Primitive of $\map \ln {x^2 - a^2}$