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Establish the integration formula for arccot x

by RoRi

Establish that the following integration formula is correct:

    \[ \int \operatorname{arccot} x \, dx = x \operatorname{arccot} x + \frac{1}{2} \log (1+x^2) + C. \]

Proof. We can establish this formula using integration by parts. Let

    \begin{align*} u &= \operatorname{arccot} x & du &= -\frac{1}{1+x^2} \\ dv &= dx & v &= x \end{align*}

where we established the formula for the derivative of \operatorname{arccot} in this exercise (Section 6.22, Exercise #3). Then we have

    \begin{align*} \int \operatorname{arccot} x \, dx &= x \operatorname{arccot} x + \int \frac{x}{1+x^2} \, dx \\ &= x \operatorname{arccot} x + \frac{1}{2} \log (1+x^2) + C. \qquad \blacksquare \end{align*}

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