What is the integral from 0 to 0.1 of xarctan(2x) ?

The integral from 0 to 0.1 of xarctan(2x) is 0.00066…

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integral from 0 to 0.1 of xarctan(2x)

\int_{0}^{0.1} x arctan (2 x) d x

0.00066 \dots

Solution steps

\int_{0}^{0.1} x arctan (2 x) d x

Apply Integration By Parts

: [\frac{1}{2} (x^{2} arctan (2 x) - 2 \cdot \int \frac{x ^{2}}{4 x ^{2} + 1} d x)]_{0}^{0.1}

= [\frac{1}{2} (x^{2} arctan (2 x) - 2 \cdot \int \frac{x ^{2}}{4 x ^{2} + 1} d x)]_{0}^{0.1}

\int \frac{x ^{2}}{4 x ^{2} + 1} d x = \frac{1}{8} (- arctan (2 x) + 2 x)

= [\frac{1}{2} (x^{2} arctan (2 x) - 2 \cdot \frac{1}{8} (- arctan (2 x) + 2 x))]_{0}^{0.1}

Simplify

[\frac{1}{2} (x^{2} arctan (2 x) - 2 \cdot \frac{1}{8} (- arctan (2 x) + 2 x))]_{0}^{0.1} : [0.5 (x^{2} arctan (2 x) - 0.25 (- arctan (2 x) + 2 x))]_{0}^{0.1}

= [0.5 (x^{2} arctan (2 x) - 0.25 (- arctan (2 x) + 2 x))]_{0}^{0.1}

Compute the boundaries:

0.00066 \dots

= 0.00066 \dots

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What is the integral from 0 to 0.1 of xarctan(2x) ?
The integral from 0 to 0.1 of xarctan(2x) is 0.00066…