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

The integral from 0 to 1 of (arctan(2x)) is arctan(2)-1/4 ln(5)

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

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

arctan (2) - \frac{1}{4} ln (5)

Decimal

0.70478 \dots

Solution steps

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

Apply Integration By Parts

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

\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}^{1}

Compute the boundaries:

arctan (2) - \frac{1}{4} ln (5)

= arctan (2) - \frac{1}{4} ln (5)

What is the integral from 0 to 1 of (arctan(2x)) ?
The integral from 0 to 1 of (arctan(2x)) is arctan(2)-1/4 ln(5)