What is the integral from-pi to 0 of-x^2-pix ?

The integral from-pi to 0 of-x^2-pix is (pi^3)/6

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integral from-pi to 0 of-x^2-pix

\int_{- π}^{0} - x^{2} - π x d x

\frac{π ^{3}}{6}

Decimal Notation

5.16771 \dots

Solution steps

\int_{- π}^{0} - x^{2} - π x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int_{- π}^{0} x^{2} d x - \int_{- π}^{0} π x d x

\int_{- π}^{0} x^{2} d x = \frac{π ^{3}}{3}

\int_{- π}^{0} π x d x = - \frac{π ^{3}}{2}

= - \frac{π ^{3}}{3} - (- \frac{π ^{3}}{2})

Simplify

- \frac{π ^{3}}{3} - (- \frac{π ^{3}}{2}) : \frac{π ^{3}}{6}

= \frac{π ^{3}}{6}

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What is the integral from-pi to 0 of-x^2-pix ?
The integral from-pi to 0 of-x^2-pix is (pi^3)/6