What is the integral from 0 to 0.1 of xarctan(x) ?

The integral from 0 to 0.1 of xarctan(x) is 0.00033…

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integral from 0 to 0.1 of xarctan(x)

\int_{0}^{0.1} x arctan (x) d x

0.00033 \dots

Solution steps

\int_{0}^{0.1} x arctan (x) d x

Apply Integration By Parts

: [\frac{1}{2} (x^{2} arctan (x) - 2 \cdot \int \frac{x ^{2}}{2 ( x ^{2} + 1 )} d x)]_{0}^{0.1}

= [\frac{1}{2} (x^{2} arctan (x) - 2 \cdot \int \frac{x ^{2}}{2 ( x ^{2} + 1 )} d x)]_{0}^{0.1}

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

= [\frac{1}{2} (x^{2} arctan (x) - 2 \cdot \frac{1}{2} (- arctan (x) + x))]_{0}^{0.1}

Simplify

[\frac{1}{2} (x^{2} arctan (x) - 2 \cdot \frac{1}{2} (- arctan (x) + x))]_{0}^{0.1} : [\frac{1}{2} (x^{2} arctan (x) + arctan (x) - x)]_{0}^{0.1}

= [\frac{1}{2} (x^{2} arctan (x) + arctan (x) - x)]_{0}^{0.1}

Compute the boundaries:

0.00033 \dots

= 0.00033 \dots

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What is the integral from 0 to 0.1 of xarctan(x) ?
The integral from 0 to 0.1 of xarctan(x) is 0.00033…