What is the integral from-pi to pi of x^3sin(x) ?

The integral from-pi to pi of x^3sin(x) is 2pi^3-12pi

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integral from-pi to pi of x^3sin(x)

\int_{- π}^{π} x^{3} sin (x) d x

2 π^{3} - 12 π

Decimal

24.31344 \dots

Solution steps

\int_{- π}^{π} x^{3} sin (x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Compute the boundaries:

2 π^{3} - 12 π

= 2 π^{3} - 12 π

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\int_{- π}^{π} x \cdot e^{- in x} d x

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