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

解

\int_{0}^{π} (x - sin^{2} (x)) d x

\frac{π ^{2} - π}{2}

十進法表記

3.36400 \dots

解答ステップ

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

\int_{0}^{π} x d x = \frac{π ^{2}}{2}

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

= \frac{π ^{2}}{2} - \frac{π}{2}

分母が等しいので, 分数を組み合わせる:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π ^{2} - π}{2}