What is the integral from 0 to 1 of xarcsin(x) ?

The integral from 0 to 1 of xarcsin(x) is pi/8

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integral from 0 to 1 of xarcsin(x)

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

\frac{π}{8}

Decimal

0.39269 \dots

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{x ^{2}}{2 1 - x ^{2}} d x = \frac{1}{4} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x)))

= [\frac{1}{2} x^{2} arcsin (x) - \frac{1}{4} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x)))]_{0}^{1}

Compute the boundaries:

\frac{π}{8}

= \frac{π}{8}

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\int_{0}^{\frac{π}{4}} sin (x) cos (x) d x

\int_{0}^{\frac{π}{2}} x^{2} cos (x) d x

\int_{0}^{8} \frac{1}{2 ( x ^{\frac{1}{3}} )} d x

What is the integral from 0 to 1 of xarcsin(x) ?
The integral from 0 to 1 of xarcsin(x) is pi/8