What is the integral from 0 to 1/2 of arctan(2x) ?

The integral from 0 to 1/2 of arctan(2x) is (pi-2ln(2))/8

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from 0 to 1/2 of arctan(2x)

\int_{0}^{\frac{1}{2}} arctan (2 x) d x

\frac{π - 2 ln ( 2 )}{8}

Decimal

0.21941 \dots

Solution steps

\int_{0}^{\frac{1}{2}} arctan (2 x) d x

Apply Integration By Parts

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

\int \frac{2 x}{4 x ^{2} + 1} d x = \frac{1}{4} ln 4 x^{2} + 1

= [x arctan (2 x) - \frac{1}{4} ln 4 x^{2} + 1]_{0}^{\frac{1}{2}}

Simplify

[x arctan (2 x) - \frac{1}{4} ln 4 x^{2} + 1]_{0}^{\frac{1}{2}} : [\frac{1}{4} (4 x arctan (2 x) - ln 4 x^{2} + 1)]_{0}^{\frac{1}{2}}

= [\frac{1}{4} (4 x arctan (2 x) - ln 4 x^{2} + 1)]_{0}^{\frac{1}{2}}

Compute the boundaries:

\frac{π - 2 ln ( 2 )}{8}

= \frac{π - 2 ln ( 2 )}{8}

What is the integral from 0 to 1/2 of arctan(2x) ?
The integral from 0 to 1/2 of arctan(2x) is (pi-2ln(2))/8