integral of 8x^3(x^4+7)^2

解

\int 8 x^{3} (x^{4} + 7)^{2} d x

8 (\frac{x ^{12}}{12} + \frac{7 x ^{8}}{4} + \frac{49 x ^{4}}{4}) + C

解答ステップ

\int 8 x^{3} (x^{4} + 7)^{2} d x

定数を除く:

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

= 8 \cdot \int x^{3} (x^{4} + 7)^{2} d x

拡張

x^{3} (x^{4} + 7)^{2} : x^{11} + 14 x^{7} + 49 x^{3}

= 8 \cdot \int x^{11} + 14 x^{7} + 49 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

= 8 (\int x^{11} d x + \int 14 x^{7} d x + \int 49 x^{3} d x)

\int x^{11} d x = \frac{x ^{12}}{12}

\int 14 x^{7} d x = \frac{7 x ^{8}}{4}

\int 49 x^{3} d x = \frac{49 x ^{4}}{4}

= 8 (\frac{x ^{12}}{12} + \frac{7 x ^{8}}{4} + \frac{49 x ^{4}}{4})

定数を解答に追加する

= 8 (\frac{x ^{12}}{12} + \frac{7 x ^{8}}{4} + \frac{49 x ^{4}}{4}) + C