What is the integral from 0 to 1/5 of yarctan(5y) ?

The integral from 0 to 1/5 of yarctan(5y) is (2pi-4)/(200)

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integral from 0 to 1/5 of yarctan(5y)

\int_{0}^{\frac{1}{5}} y arctan (5 y) d y

\frac{2 π - 4}{200}

Decimal Notation

0.01141 \dots

Solution steps

\int_{0}^{\frac{1}{5}} y arctan (5 y) d y

Apply Integration By Parts

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

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

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

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

2 \cdot \frac{1}{50} (- arctan (5 y) + 5 y) = \frac{1}{25} (- arctan (5 y) + 5 y)

= [\frac{1}{2} (y^{2} arctan (5 y) - \frac{1}{25} (- arctan (5 y) + 5 y))]_{0}^{\frac{1}{5}}

Compute the boundaries:

\frac{2 π - 4}{200}

= \frac{2 π - 4}{200}

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\int_{2}^{6} (3 x + 4) d x

What is the integral from 0 to 1/5 of yarctan(5y) ?
The integral from 0 to 1/5 of yarctan(5y) is (2pi-4)/(200)