integral of (7x^3-5)^2

解

\int (7 x^{3} - 5)^{2} d x

7 x^{7} - \frac{35 x ^{4}}{2} + 25 x + C

解答ステップ

\int (7 x^{3} - 5)^{2} d x

拡張

(7 x^{3} - 5)^{2} : 49 x^{6} - 70 x^{3} + 25

= \int 49 x^{6} - 70 x^{3} + 25 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 49 x^{6} d x - \int 70 x^{3} d x + \int 25 d x

\int 49 x^{6} d x = 7 x^{7}

\int 70 x^{3} d x = \frac{35 x ^{4}}{2}

\int 25 d x = 25 x

= 7 x^{7} - \frac{35 x ^{4}}{2} + 25 x

定数を解答に追加する

= 7 x^{7} - \frac{35 x ^{4}}{2} + 25 x + C