What is the integral from 0 to 0.1 of xarctan(5x) ?

The integral from 0 to 0.1 of xarctan(5x) is 0.00159…

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integral from 0 to 0.1 of xarctan(5x)

\int_{0}^{0.1} x arctan (5 x) d x

0.00159 \dots

Solution steps

\int_{0}^{0.1} x arctan (5 x) d x

Apply Integration By Parts

: [\frac{1}{2} (x^{2} arctan (5 x) - 2 \cdot \int \frac{5 x ^{2}}{2 ( 25 x ^{2} + 1 )} d x)]_{0}^{0.1}

= [\frac{1}{2} (x^{2} arctan (5 x) - 2 \cdot \int \frac{5 x ^{2}}{2 ( 25 x ^{2} + 1 )} d x)]_{0}^{0.1}

\int \frac{5 x ^{2}}{2 ( 25 x ^{2} + 1 )} d x = \frac{1}{50} (- arctan (5 x) + 5 x)

= [\frac{1}{2} (x^{2} arctan (5 x) - 2 \cdot \frac{1}{50} (- arctan (5 x) + 5 x))]_{0}^{0.1}

Simplify

[\frac{1}{2} (x^{2} arctan (5 x) - 2 \cdot \frac{1}{50} (- arctan (5 x) + 5 x))]_{0}^{0.1} : [0.5 (x^{2} arctan (5 x) - 0.04 (- arctan (5 x) + 5 x))]_{0}^{0.1}

= [0.5 (x^{2} arctan (5 x) - 0.04 (- arctan (5 x) + 5 x))]_{0}^{0.1}

Compute the boundaries:

0.00159 \dots

= 0.00159 \dots

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\frac{d}{d x} (1 + 7, x - tan (x))

What is the integral from 0 to 0.1 of xarctan(5x) ?
The integral from 0 to 0.1 of xarctan(5x) is 0.00159…