integral from-c to c of (c-x^2)-(x^2-c)

解

\int_{- c}^{c} (c - x^{2}) - (x^{2} - c) d x

- \frac{4 c ^{3}}{3} + 4 c^{2}

解答ステップ

\int_{- c}^{c} c - x^{2} - (x^{2} - c) 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_{- c}^{c} c d x - \int_{- c}^{c} x^{2} d x - \int_{- c}^{c} x^{2} - c d x

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

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

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

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

簡素化

2 c^{2} - \frac{2 c ^{3}}{3} - (\frac{2 c ^{3}}{3} - 2 c^{2}) : - \frac{4 c ^{3}}{3} + 4 c^{2}

= - \frac{4 c ^{3}}{3} + 4 c^{2}