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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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 Notation

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}

= [\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}

n = 1 \sum \infty \frac{10}{3 ^{n}}

\frac{d}{d x} (sec^{2} (x))

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

\int \frac{x ^{2} - a ^{2}}{x} d x

\frac{\partial}{\partial y} (cos (x^{2}) sin (yz))

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