What is the integral from-1 to 1 of xarctan(x) ?

The integral from-1 to 1 of xarctan(x) is (2pi-4)/4

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integral from-1 to 1 of xarctan(x)

\int_{- 1}^{1} x arctan (x) d x

\frac{2 π - 4}{4}

Decimal

0.57079 \dots

Solution steps

\int_{- 1}^{1} x arctan (x) d x

Apply Integration By Parts

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

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

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

Simplify

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

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

Compute the boundaries:

\frac{2 π - 4}{4}

= \frac{2 π - 4}{4}

What is the integral from-1 to 1 of xarctan(x) ?
The integral from-1 to 1 of xarctan(x) is (2pi-4)/4