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

解

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

\frac{54700191}{16}

十進法表記

3418761.9375

解答ステップ

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

拡張

(x^{4} + 5)^{3} x^{3} : x^{15} + 15 x^{11} + 75 x^{7} + 125 x^{3}

= \int_{0}^{3} x^{15} + 15 x^{11} + 75 x^{7} + 125 x^{3} 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}^{3} x^{15} d x + \int_{0}^{3} 15 x^{11} d x + \int_{0}^{3} 75 x^{7} d x + \int_{0}^{3} 125 x^{3} d x

\int_{0}^{3} x^{15} d x = \frac{43046721}{16}

\int_{0}^{3} 15 x^{11} d x = \frac{2657205}{4}

\int_{0}^{3} 75 x^{7} d x = \frac{492075}{8}

\int_{0}^{3} 125 x^{3} d x = \frac{10125}{4}

= \frac{43046721}{16} + \frac{2657205}{4} + \frac{492075}{8} + \frac{10125}{4}

簡素化

\frac{43046721}{16} + \frac{2657205}{4} + \frac{492075}{8} + \frac{10125}{4} : \frac{54700191}{16}

= \frac{54700191}{16}