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

The integral from-1 to 1 of xarcsin(x) is pi/4

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integral from-1 to 1 of xarcsin(x)

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

\frac{π}{4}

Decimal

0.78539 \dots

Solution steps

\int_{- 1}^{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]_{- 1}^{1}

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

Compute the boundaries:

\frac{π}{4}

= \frac{π}{4}

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What is the integral from-1 to 1 of xarcsin(x) ?
The integral from-1 to 1 of xarcsin(x) is pi/4