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

The integral from 0 to 1 of x^2arcsin(x) is pi/6-2/9

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

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

\frac{π}{6} - \frac{2}{9}

Solution steps

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

Apply Integration By Parts

: [\frac{1}{3} x^{3} arcsin (x) - \int \frac{x ^{3}}{3 1 - x ^{2}} d x]_{0}^{1}

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

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

= [\frac{1}{3} x^{3} arcsin (x) - \frac{1}{3} (\frac{1}{3} (1 - x^{2})^{\frac{3}{2}} - 1 - x^{2})]_{0}^{1}

Compute the boundaries:

\frac{π}{6} - \frac{2}{9}

= \frac{π}{6} - \frac{2}{9}

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

\int_{2}^{3} 4 - x^{2} d x

\int_{- 1}^{1} x^{2} - x d x

\int_{1}^{2} \frac{1}{x + 1} d x

What is the integral from 0 to 1 of x^2arcsin(x) ?
The integral from 0 to 1 of x^2arcsin(x) is pi/6-2/9