What is the integral from 0 to pi of x^2*sin(x) ?

The integral from 0 to pi of x^2*sin(x) is pi^2-4

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integral from 0 to pi of x^2*sin(x)

\int_{0}^{π} x^{2} \cdot sin (x) d x

π^{2} - 4

Decimal

5.86960 \dots

Solution steps

\int_{0}^{π} x^{2} sin (x) d x

Apply Integration By Parts

: [- x^{2} cos (x) - \int - 2 x cos (x) d x]_{0}^{π}

= [- x^{2} cos (x) - \int - 2 x cos (x) d x]_{0}^{π}

\int - 2 x cos (x) d x = - 2 (x sin (x) + cos (x))

= [- x^{2} cos (x) - (- 2 (x sin (x) + cos (x)))]_{0}^{π}

Simplify

= [- x^{2} cos (x) + 2 (x sin (x) + cos (x))]_{0}^{π}

Compute the boundaries:

π^{2} - 4

= π^{2} - 4

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What is the integral from 0 to pi of x^2*sin(x) ?
The integral from 0 to pi of x^2*sin(x) is pi^2-4