Primitive of x by Arctangent of x over a

Theorem

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

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

let:

and let:

Then:

Also see

 * Primitive of $x \arcsin \dfrac x a$


 * Primitive of $x \arccos \dfrac x a$


 * Primitive of $x \arccot \dfrac x a$


 * Primitive of $x \arcsec \dfrac x a$


 * Primitive of $x \arccsc \dfrac x a$