What is the integral from 0 to 1/5 of arctan(5x) ?

The integral from 0 to 1/5 of arctan(5x) is (pi-2ln(2))/(20)

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integral from 0 to 1/5 of arctan(5x)

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

\frac{π - 2 ln ( 2 )}{20}

Decimal Notation

0.08776 \dots

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{5 x}{25 x ^{2} + 1} d x = \frac{1}{10} ln 25 x^{2} + 1

= [x arctan (5 x) - \frac{1}{10} ln 25 x^{2} + 1]_{0}^{\frac{1}{5}}

Simplify

[x arctan (5 x) - \frac{1}{10} ln 25 x^{2} + 1]_{0}^{\frac{1}{5}} : [\frac{1}{10} (10 x arctan (5 x) - ln 25 x^{2} + 1)]_{0}^{\frac{1}{5}}

= [\frac{1}{10} (10 x arctan (5 x) - ln 25 x^{2} + 1)]_{0}^{\frac{1}{5}}

Compute the boundaries:

\frac{π - 2 ln ( 2 )}{20}

= \frac{π - 2 ln ( 2 )}{20}

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What is the integral from 0 to 1/5 of arctan(5x) ?
The integral from 0 to 1/5 of arctan(5x) is (pi-2ln(2))/(20)