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

The integral from 0 to 1 of x^2arctan(x) is pi/(12)+(ln(2)-1)/6

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

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

\frac{π}{12} + \frac{ln ( 2 ) - 1}{6}

Decimal

0.21065 \dots

Solution steps

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

Apply Integration By Parts

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

\int \frac{x ^{3}}{3 ( x ^{2} + 1 )} d x = \frac{1}{6} (x^{2} + 1 - ln x^{2} + 1)

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

Compute the boundaries:

\frac{π}{12} - \frac{- ln ( 2 ) + 2}{6} + \frac{1}{6}

= \frac{π}{12} - \frac{- ln ( 2 ) + 2}{6} + \frac{1}{6}

Simplify

= \frac{π}{12} + \frac{ln ( 2 ) - 1}{6}

What is the integral from 0 to 1 of x^2arctan(x) ?
The integral from 0 to 1 of x^2arctan(x) is pi/(12)+(ln(2)-1)/6