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

The integral from 0 to 1 of xarctan(x^2) is pi/8-1/4 ln(2)

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integral from 0 to 1 of xarctan(x^2)

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

\frac{π}{8} - \frac{1}{4} ln (2)

Decimal

0.21941 \dots

Solution steps

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

Apply Integration By Parts

= [\frac{1}{2} x^{2} arctan (x^{2}) - \int \frac{x ^{3}}{x ^{4} + 1} d x]_{0}^{1}

\int \frac{x ^{3}}{x ^{4} + 1} d x = \frac{1}{4} ln x^{4} + 1

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

Compute the boundaries:

\frac{π}{8} - \frac{1}{4} ln (2)

= \frac{π}{8} - \frac{1}{4} ln (2)

What is the integral from 0 to 1 of xarctan(x^2) ?
The integral from 0 to 1 of xarctan(x^2) is pi/8-1/4 ln(2)