What is the integral from 0 to pi of x^3cos(x) ?

The integral from 0 to pi of x^3cos(x) is -3pi^2+12

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integral from 0 to pi of x^3cos(x)

\int_{0}^{π} x^{3} cos (x) d x

- 3 π^{2} + 12

Decimal

- 17.60881 \dots

Solution steps

\int_{0}^{π} x^{3} cos (x) d x

Apply Integration By Parts

: [x^{3} sin (x) - \int 3 x^{2} sin (x) d x]_{0}^{π}

= [x^{3} sin (x) - \int 3 x^{2} sin (x) d x]_{0}^{π}

\int 3 x^{2} sin (x) d x = 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x)))

= [x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x)))]_{0}^{π}

Compute the boundaries:

- 3 π^{2} + 12

= - 3 π^{2} + 12

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What is the integral from 0 to pi of x^3cos(x) ?
The integral from 0 to pi of x^3cos(x) is -3pi^2+12