What is the integral from 0 to 1 of xarccot(x) ?

The integral from 0 to 1 of xarccot(x) is 1/2

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integral from 0 to 1 of xarccot(x)

\int_{0}^{1} x arccot (x) d x

\frac{1}{2}

Decimal Notation

0.5

Solution steps

\int_{0}^{1} x arccot (x) d x

Apply Integration By Parts

: [\frac{1}{2} x^{2} arccot (x) - \int - \frac{x ^{2}}{2 ( x ^{2} + 1 )} d x]_{0}^{1}

= [\frac{1}{2} x^{2} arccot (x) - \int - \frac{x ^{2}}{2 ( x ^{2} + 1 )} d x]_{0}^{1}

\int - \frac{x ^{2}}{2 ( x ^{2} + 1 )} d x = - \frac{1}{2} (- arctan (x) + x)

= [\frac{1}{2} x^{2} arccot (x) - (- \frac{1}{2} (- arctan (x) + x))]_{0}^{1}

Simplify

[\frac{1}{2} x^{2} arccot (x) - (- \frac{1}{2} (- arctan (x) + x))]_{0}^{1} : [\frac{1}{2} (x^{2} arccot (x) - arctan (x) + x)]_{0}^{1}

= [\frac{1}{2} (x^{2} arccot (x) - arctan (x) + x)]_{0}^{1}

Compute the boundaries:

\frac{1}{2}

= \frac{1}{2}

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What is the integral from 0 to 1 of xarccot(x) ?
The integral from 0 to 1 of xarccot(x) is 1/2