integral from 0 to 2 of (x^4+5)^3

解

\int_{0}^{2} (x^{4} + 5)^{3} d x

\frac{86326}{39}

十進法表記

2213.48717 \dots

解答ステップ

\int_{0}^{2} (x^{4} + 5)^{3} d x

拡張

(x^{4} + 5)^{3} : x^{12} + 15 x^{8} + 75 x^{4} + 125

= \int_{0}^{2} x^{12} + 15 x^{8} + 75 x^{4} + 125 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}^{2} x^{12} d x + \int_{0}^{2} 15 x^{8} d x + \int_{0}^{2} 75 x^{4} d x + \int_{0}^{2} 125 d x

\int_{0}^{2} x^{12} d x = \frac{8192}{13}

\int_{0}^{2} 15 x^{8} d x = \frac{2560}{3}

\int_{0}^{2} 75 x^{4} d x = 480

\int_{0}^{2} 125 d x = 250

= \frac{8192}{13} + \frac{2560}{3} + 480 + 250

簡素化

\frac{8192}{13} + \frac{2560}{3} + 480 + 250 : \frac{86326}{39}

= \frac{86326}{39}