integral from-1 to 1 of k(1-x^2)

解

\int_{- 1}^{1} k (1 - x^{2}) d x

\frac{4}{3} k

解答ステップ

\int_{- 1}^{1} k (1 - x^{2}) d x

定数を除く:

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

= k \cdot \int_{- 1}^{1} 1 - x^{2} 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

= k (\int_{- 1}^{1} 1 d x - \int_{- 1}^{1} x^{2} d x)

\int_{- 1}^{1} 1 d x = 2

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

= k (2 - \frac{2}{3})

結合

2 - \frac{2}{3} : \frac{4}{3}

= \frac{4}{3} k