integral of 30(x+7)(x+5)^4

解

\int 30 (x + 7) (x + 5)^{4} d x

30 (\frac{x ^{6}}{6} + \frac{27 x ^{5}}{5} + \frac{145 x ^{4}}{2} + \frac{1550 x ^{3}}{3} + \frac{4125 x ^{2}}{2} + 4375 x) + C

解答ステップ

\int 30 (x + 7) (x + 5)^{4} d x

定数を除く:

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

= 30 \cdot \int (x + 7) (x + 5)^{4} d x

拡張

(x + 7) (x + 5)^{4} : x^{5} + 27 x^{4} + 290 x^{3} + 1550 x^{2} + 4125 x + 4375

= 30 \cdot \int x^{5} + 27 x^{4} + 290 x^{3} + 1550 x^{2} + 4125 x + 4375 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

= 30 (\int x^{5} d x + \int 27 x^{4} d x + \int 290 x^{3} d x + \int 1550 x^{2} d x + \int 4125 x d x + \int 4375 d x)

\int x^{5} d x = \frac{x ^{6}}{6}

\int 27 x^{4} d x = \frac{27 x ^{5}}{5}

\int 290 x^{3} d x = \frac{145 x ^{4}}{2}

\int 1550 x^{2} d x = \frac{1550 x ^{3}}{3}

\int 4125 x d x = \frac{4125 x ^{2}}{2}

\int 4375 d x = 4375 x

= 30 (\frac{x ^{6}}{6} + \frac{27 x ^{5}}{5} + \frac{145 x ^{4}}{2} + \frac{1550 x ^{3}}{3} + \frac{4125 x ^{2}}{2} + 4375 x)

定数を解答に追加する

= 30 (\frac{x ^{6}}{6} + \frac{27 x ^{5}}{5} + \frac{145 x ^{4}}{2} + \frac{1550 x ^{3}}{3} + \frac{4125 x ^{2}}{2} + 4375 x) + C