Primitive of Arccotangent of x over a over x

Theorem

$\ds \int \dfrac 1 x \arccot \frac x a \rd x = \frac \pi 2 \ln \size x - \int \dfrac 1 x \arctan \frac x a \rd x$

Proof

\(\ds \int \dfrac 1 x \arccot \frac x a \rd x\)

\(=\)

\(\ds \int \dfrac 1 x \paren {\frac \pi 2 - \arctan \frac x a} \rd x\)

Sum of Arctangent and Arccotangent

\(\ds \)

\(=\)

\(\ds \frac \pi 2 \int \frac {\d x} x - \int \dfrac 1 x \arctan \frac x a \rd x\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds \frac \pi 2 \ln \size x - \int \dfrac 1 x \arctan \frac x a \rd x + C\)

Primitive of Reciprocal

$\blacksquare$

Also see

• Primitive of $\dfrac 1 x \arcsin \dfrac x a$

• Primitive of $\dfrac 1 x \arccos \dfrac x a$

• Primitive of $\dfrac 1 x \arctan \dfrac x a$

• Primitive of $\dfrac 1 x \arcsec \dfrac x a$

• Primitive of $\dfrac 1 x \arccsc \dfrac x a$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.491$