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

The integral from 0 to 1 of (arccos(x))^3 is 3((pi^2)/4-2)

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

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

3 (\frac{π ^{2}}{4} - 2)

Decimal Notation

1.40220 \dots

Solution steps

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

Simplify

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

Apply Integration By Parts

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

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

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

= [x arccos^{3} (x) - 3 (arccos^{2} (x) 1 - x^{2} - 2 (- x arccos (x) + 1 - x^{2}))]_{0}^{1}

Compute the boundaries:

3 (\frac{π ^{2}}{4} - 2)

= 3 (\frac{π ^{2}}{4} - 2)

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What is the integral from 0 to 1 of (arccos(x))^3 ?
The integral from 0 to 1 of (arccos(x))^3 is 3((pi^2)/4-2)