What is the integral from 0 to 1/3 of arctan(3x) ?

The integral from 0 to 1/3 of arctan(3x) is (pi-2ln(2))/(12)

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integral from 0 to 1/3 of arctan(3x)

\int_{0}^{\frac{1}{3}} arctan (3 x) d x

\frac{π - 2 ln ( 2 )}{12}

Decimal Notation

0.14627 \dots

Solution steps

\int_{0}^{\frac{1}{3}} arctan (3 x) d x

Apply Integration By Parts

: [x arctan (3 x) - \int \frac{3 x}{9 x ^{2} + 1} d x]_{0}^{\frac{1}{3}}

= [x arctan (3 x) - \int \frac{3 x}{9 x ^{2} + 1} d x]_{0}^{\frac{1}{3}}

\int \frac{3 x}{9 x ^{2} + 1} d x = \frac{1}{6} ln 9 x^{2} + 1

= [x arctan (3 x) - \frac{1}{6} ln 9 x^{2} + 1]_{0}^{\frac{1}{3}}

Simplify

[x arctan (3 x) - \frac{1}{6} ln 9 x^{2} + 1]_{0}^{\frac{1}{3}} : [\frac{1}{6} (6 x arctan (3 x) - ln 9 x^{2} + 1)]_{0}^{\frac{1}{3}}

= [\frac{1}{6} (6 x arctan (3 x) - ln 9 x^{2} + 1)]_{0}^{\frac{1}{3}}

Compute the boundaries:

\frac{π - 2 ln ( 2 )}{12}

= \frac{π - 2 ln ( 2 )}{12}

\int_{0}^{2} (x + 1)^{3} d x

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\int_{- 2}^{2} 6 - x^{2} - ∣ x ∣ d x

\int_{0}^{8} ∣ x - 5 ∣ d x

\int_{- 2}^{- 4} (x + 4)^{10} d x

What is the integral from 0 to 1/3 of arctan(3x) ?
The integral from 0 to 1/3 of arctan(3x) is (pi-2ln(2))/(12)