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

The integral from 0 to 1 of x^3arctan(x) is 1/6

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

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

\frac{1}{6}

Solution steps

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

Apply Integration By Parts

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

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

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

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

Compute the boundaries:

\frac{1}{6}

= \frac{1}{6}

\int_{0}^{2} x^{3} - 2 x d x

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

\int_{2}^{4} (\frac{x}{2} - \frac{2 x ^{2}}{3}) d x

\int_{- 1}^{1} x^{- 1} d x

\int_{0}^{π} x^{4} cos (2 x^{5}) d x

What is the integral from 0 to 1 of x^3arctan(x) ?
The integral from 0 to 1 of x^3arctan(x) is 1/6