integral from 0 to pi of-x^2+pix-sin(x)

解

\int_{0}^{π} - x^{2} + π x - sin (x) d x

\frac{π ^{3}}{6} - 2

十進法表記

3.16771 \dots

解答ステップ

\int_{0}^{π} - x^{2} + π x - sin (x) d x

総和規則を適用する:

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

= - \int_{0}^{π} x^{2} d x + \int_{0}^{π} π x d x - \int_{0}^{π} sin (x) d x

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

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

\int_{0}^{π} sin (x) d x = 2

= - \frac{π ^{3}}{3} + \frac{π ^{3}}{2} - 2

分数を組み合わせる

- \frac{π ^{3}}{3} + \frac{π ^{3}}{2} : \frac{π ^{3}}{6}

= \frac{π ^{3}}{6} - 2