integral of (3x+1)(2x-5)^2

解

\int (3 x + 1) (2 x - 5)^{2} d x

3 x^{4} - \frac{56 x ^{3}}{3} + \frac{55 x ^{2}}{2} + 25 x + C

解答ステップ

\int (3 x + 1) (2 x - 5)^{2} d x

拡張

(3 x + 1) (2 x - 5)^{2} : 12 x^{3} - 56 x^{2} + 55 x + 25

= \int 12 x^{3} - 56 x^{2} + 55 x + 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 12 x^{3} d x - \int 56 x^{2} d x + \int 55 x d x + \int 25 d x

\int 12 x^{3} d x = 3 x^{4}

\int 56 x^{2} d x = \frac{56 x ^{3}}{3}

\int 55 x d x = \frac{55 x ^{2}}{2}

\int 25 d x = 25 x

= 3 x^{4} - \frac{56 x ^{3}}{3} + \frac{55 x ^{2}}{2} + 25 x

定数を解答に追加する

= 3 x^{4} - \frac{56 x ^{3}}{3} + \frac{55 x ^{2}}{2} + 25 x + C