Primitive of Power of x by Arctangent of x over a

Theorem

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

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 $x^m \arcsin \dfrac x a$


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


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


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


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