integral from 0 to 1 of pi(x^2+1)

解

\int_{0}^{1} π (x^{2} + 1) d x

π \frac{4}{3}

十進法表記

4.18879 \dots

解答ステップ

\int_{0}^{1} π (x^{2} + 1) d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= π \cdot \int_{0}^{1} x^{2} + 1 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

= π (\int_{0}^{1} x^{2} d x + \int_{0}^{1} 1 d x)

\int_{0}^{1} x^{2} d x = \frac{1}{3}

\int_{0}^{1} 1 d x = 1

= π (\frac{1}{3} + 1)

結合

\frac{1}{3} + 1 : \frac{4}{3}

= π \frac{4}{3}