Primitive of Arctangent Function

Theorem

$\ds \int \arctan x \rd x = x \arctan x - \frac {\map \ln {x^2 + 1} } 2 + C$

Proof

From Primitive of $\arctan \dfrac x a$:

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

The result follows by setting $a = 1$.

$\blacksquare$

Also presented as

This result can also be presented as:

$\ds \int \arctan x \rd x = x \arctan x - \ln \sqrt {x^2 + 1} + C$

Also see

• Primitive of $\arcsin x$
• Primitive of $\arccos x$
• Primitive of $\arccot x$

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): arc-tangent
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals