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

解

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

2 a^{5} - \frac{4 a ^{5}}{3} + \frac{2 a ^{5}}{5}

解答ステップ

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

拡張

(a^{2} - x^{2})^{2} : a^{4} - 2 a^{2} x^{2} + x^{4}

= \int_{- a}^{a} a^{4} - 2 a^{2} x^{2} + x^{4} 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_{- a}^{a} a^{4} d x - \int_{- a}^{a} 2 a^{2} x^{2} d x + \int_{- a}^{a} x^{4} d x

\int_{- a}^{a} a^{4} d x = 2 a^{5}

\int_{- a}^{a} 2 a^{2} x^{2} d x = \frac{4 a ^{5}}{3}

\int_{- a}^{a} x^{4} d x = \frac{2 a ^{5}}{5}

= 2 a^{5} - \frac{4 a ^{5}}{3} + \frac{2 a ^{5}}{5}