integral of x(5x-3)^2

解

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

\frac{25 x ^{4}}{4} - 10 x^{3} + \frac{9 x ^{2}}{2} + C

解答ステップ

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

因数

5 x - 3 : - (- 5 x + 3)

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

(- (- 5 x + 3))^{2} = (- 5 x + 3)^{2}

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

拡張

x (- 5 x + 3)^{2} : 25 x^{3} - 30 x^{2} + 9 x

= \int 25 x^{3} - 30 x^{2} + 9 x 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 25 x^{3} d x - \int 30 x^{2} d x + \int 9 x d x

\int 25 x^{3} d x = \frac{25 x ^{4}}{4}

\int 30 x^{2} d x = 10 x^{3}

\int 9 x d x = \frac{9 x ^{2}}{2}

= \frac{25 x ^{4}}{4} - 10 x^{3} + \frac{9 x ^{2}}{2}

定数を解答に追加する

= \frac{25 x ^{4}}{4} - 10 x^{3} + \frac{9 x ^{2}}{2} + C